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Craftsmanship in the 

Teaching of Elementary 

Mathematics 



BLACKIE & SON LIMITED 

50 Old Bailey, LONDON 
17 Stanhope Street, GLASGOW 

BLACKIE & SON (INDIA) LIMITED 
Warwick House, Fort Street, BOMBAY 

BLACKIE & SON (CANADA) LIMITED 
TORONTO 



Craftsmanship in_the_ 
Teaching of Elementary 

^ - - -O - " v 

Mathematics 



BY 



F. W. WESTAWAY 



Formerly one of H. M. Inspectors of Secondary Schools 
Author of " Scientific Method, its Philosophical Basis and its Modes of A 
" Science Teaching: What it Was What it Is What it Might E 
" The Endless Quest: 3000 Years of Science " &c. 



BLACKIE & SON LIMITED 

LONDON AND GLASGOW 



First issued 1931 
Reprinted 1934, *937 



Printed in Great Britain by Blackie & Son, Ltd., Glasgow 



" Have some wine" the March Hare said in an 
encouraging tone. 

Alice looked round the table, but there was nothing 
on it but tea. " / don't see any wine" she remarked. 

" There isn't any'' said the March Hare. 

" Then it wasn't very civil of you to offer it" 
said Alice angrily. 



" You are sad," the White Knight said. " Let me 
sing you a song to comfort you." 

" Is it very long?" Alice asked, for she had heard a 
good deal of poetry that day. 

" It's long" said the Knight, " but it's very, very 
beautiful. Everybody that hears me sing it either it 
brings the tears into their eyes, or else " 

" Or else what?" said Alice, for the Knight had 
made a pause. 

" Or else it doesn't, you know. The song is called 
' WAYS AND MEANS ', but that's only what it's called, 
you know!" 

" Well, what is the song, then?" said Alice. 

" / was coming to that," the Knight said. " The 
song really is ' A-SITTING ON A GATE ': and the tune's 
my own invention" 



BY F. W. WESTAWAY 

SCIENCE TEACHING 

What it Was What it Is 

What it Might Be 
Second Impression. 10s. 6d. net 

" Get the book and read it; it is the best thing yet. It is packed 
with practical advice which will always be of value." 

Journal of Education. 

" His hook wll set many a young teacher on the right path, and 
will help many an older one to raise his performance to a much 
higher level or excellence." Nature. 

"Reveals on every page the zestful interest of a true craftsman in 
teaching blended with informed good sense. . . . This book should be 
read by all headmasters and headmistresses in secondary schools, and 
it is worthy to be studied by every teacher of science. If its counsels 
are adopted and followed we shall see a great and beneficent change 
in the present method of dealing with science as a factor in education." 

Education Outlook. 

" This is a remarkable book, critical and stimulating, the product of 
the author's long experience as teacher, headmaster, and H.M.I. 
. . . comprehensive in scope and so practical that it will be a most 
helpful guide to the beginner and an inspiration to all. We recommend 
it unreservedly to all engaged in science teaching in schools and 
universities." School Science Review. 



PREFACE 



When asked to write a book on the teaching of Elementary 
Mathematics, I felt doubtful as to the avenue by which the 
subject might be best approached. During the present cen- 
tury, the general " policy " and " attitude " to be adopted in 
mathematical teaching have been discussed by so many 
authorities that there seemed very little new to say. Finally 
I decided that class-room craftsmanship might be made a 
suitable basis of treatment. Thus the book is not intended for 
the experienced teacher who has already acquired skill in his 
art, but for the still struggling beginner. In the leading schools, 
mathematical craftsmanship probably leaves little to be desired, 
but the leaven has yet to work its way into the mass. 

From the great variety of topics that come within the 
ambit of the various mathematical subjects, I have selected 
for treatment those which, in my experience, seem to give 
young teachers most difficulty. To treat all topics that come 
within the daily practice of mathematical teachers is impossible; 
it would mean writing a dozen books rather than one. 

I have sometimes been asked if, as an Inspector pursuing 
the same daily round year after year, decade after decade, I 
am, when listening to lessons in mathematics, ever amused, 
ever really interested, ever inclined to be severely critical, 
ever bored. 

vii 



viii PREFACE 

Amused? Yes; for instance, when a young master tells his 
boys that mathematics is by far the most important subject 
they learn, inasmuch as it is the only one that leads them into 
the region of " pure thought ". 

Really interested? Yes, every day of my life. In the crafts- 
manship of even a beginner there is almost always some element 
of interest; in the craftsmanship of a really skilful mathe- 
matical teacher there is to me always a veritable joy. I never 
enter a classroom without hoping to find something which will 
make an appeal, and I am not often disappointed. Sometimes 
disappointed, of course; unfortunately not all mathematical 
teachers have come down from heaven. 

Severely critical? Yes, occasionally, more especially at the 
rather slavish adoption of certain doubtful forms of traditional 
procedure. For instance, a teacher may include in the work 
of the bottom " Set " of a Form the Italian method of division, 
well knowing that two-thirds of the boys will thenceforth 
always get their sums wrong. Another teacher may adopt 
" standard form ", not because he has examined it and found 
it to be good, but because " everybody does it nowadays ". 
Instead of saying, " I thought we had to do these things," 
why do not mathematical teachers hold fast to the faith which 
is really in them? If their faith, their faith, includes the Italian 
method, standard form, and the score of other doubtful ex- 
pedients that spread like measles from school to school, I 
have nothing more to say. 

Bored? Yes, though not often. The petrifying stuff often 
doled out to Sixth Form specialists, the everlasting Series and 
Progressions, the old dodges and devices and bookwork "proofs" 
ad nauseam in preparation for scholarship examinations, all 
this is virtually the same now as forty years ago. True, teachers 



PREFACE ix 

are not much to blame for this. Boys have to be prepared for 
the scholarship examinations, and according to prescription. 
But that does not soothe an Inspector who has to listen to the 
same thing year after year, and I admit that, with Sixth Form 
work, sometimes I am almost bored to tears. 

If I had to pick out those topics which in the classroom 
make the strongest appeal to me, I should include (i) Arith- 
metic to six-, seven-, and eight-year-olds, when well taught; 
(ii) Beginners' geometry; (iii) Upper Fourth and Lower Fifth 
work when the rather more advanced topics in algebra, geo- 
metry, trigonometry, and mechanics are being taken for the first 
time (not the Upper Fifth and its revision work); (iv) Upper 
Sixth work when examinations are over and the chief mathe- 
matical master really has a chance to show himself as a master 
of his craft. Sixth and Upper Fifth Form work often savours 
too much of the examination room to be greatly interesting: 
everything is excluded that does not pay. But inasmuch as 
examination success is a question of bread and cheese to the 
boy, the teacher is really on the horns of a dilemma, and very 
naturally he prefers to transfix himself on that horn that brings 
him the less pain. 

Why, of all the subjects taught, is mathematics the least 
popular in girls' schools? and whyjs [t_the one subject in 
which the man in the street feels no personal interest? 

It is not because mathematics is difficult to teach. My own 
opinion is that it improbably the easiest of all subjects to teach. 
When it is taught by well-qualified mathematicians, and when 
those mathematicians are skilled in their teaching craft, suc- 
cess seems always to follow as a matter of course, in girls' 
schools equally with boys'. The failure to make any head- 
way, even under the best conditions, on the part of a small 



x PREFACE 

proportion of boys and a rather larger proportion of girls is 
probably due to a natural incapacity for the subject. Had 
I my own way, I would debar any teacher from teaching even 
elementary mathematics who had not taken a strong. -doge^ of 
the calculus, and covered a fairly extensive field of advanced 
work generally. It is idle to expect a mathematical teacher to 
handle even elementary mathematics properly unless he has 
begjx.through the mathematical mill. And yet I have heard 
a Headmaster say, " He can take the Lower Form mathe- 
matics all right; he is one of my useful men: he took a Third 
in History." 

As long as University Scholarships are what they are, so 
long will Sixth Form specialists* work proceed on present 
lines. But one purpose of the book is to plead for considera- 
tion of the many neglected byways in mathematics and for 
their inclusion in a course for all Sixth Form boys; suggestions 
to this end are made in some of the later chapters. We want 
a far greater number of ordinary pupils to become mathe- 
matically interested, interested in such a way that the interest 
will be permanent; and we want them to learn to think mathe- 
matically, if only in a very moderate degree. Why do ordinary 
pupils shrivel up when they find a mathematician in their 
midst? It is simply that they are afraid of his cold logic. 

There is, in fact, a curious popular prejudice against mathe- 
maticians as a class. It probably arises from the fact that we 
*~~_-.._ .~ - --., * j 

are not a nation of clear thinkers, and we dislike the few amongst 
us who are. Foreigners at least the French, the Germans, 
and the Italians are mathematically much keener than we 
are. They seem to become immediately interested in a 
topic with mathematical associations, whereas we turn away 
from it, disinclined to take part in a discussion demanding 



PREFACE xi 

rigorous logical reasoning. Competent observers agree that 
this is in no small measure traceable to the fact that our 
school mathematics has not been of a type to leave on the 
minds of ordinary pupils impressions of permanent interest. 

We^ have driven Euclid out of Britain, but we must all 
admit that he stood as a model of honest thinking, and we 
miss him sadly. Were he to come back, frankly admitting his 
failings and promising reform, not a few mathematicians 
would give him a warm welcome. It is only a very few years 
since I heard my last lesson in Euclid, and that, curiously 
enough, was at a Preparatory School. It was a pleasure to 
hear those 12-year-old boys promptly naming their authority 
(e.g. I, 32; III, 21; II, 11) for every statement they made, and it 
was exceedingly difficult to improvise the necessary frown of 
disapproval. Of course those small boys did not understand 
much of what they were doing, and of actual geometry they 
knew little. But in spite of this they were learning to think 
logically, and to produce good authority for every assertion 
they made. Our modern ways are doubtless better than the 
old ways, but when we emptied the bath, why did we throw 
out the baby? 

The very last thing I desire to do is to impose on teachers 
my ideas of methods. Anything of the nature of a standardized 
mgthpd in English schools is unthinkable. The Board of 
Education, as I knew it, never issued decrees in matters affect- 
ing the faith and doctrines of our educational system; it con- 
fined itself to making suggestions. Admittedly, however, 
democracy has now come to stay, and its unfortunate though 
inevitable tendency to standardize everything it controls may 
ultimately prove disastrous to all originality^in teaching methods, 
and reduce the past high average ofjnitm^ve^agdjrf 



xii PREFACE 

independence in schools. Let^every; teacher^strive to base his 
methods., on a venturous originality. Let him resist to the 
death all attempts of all bureaucrats to loosen the bonds of 
obligation to his art, or to mar his craftsmanship. 

But though I plead for originality I desire to utter a warn- 
ing against a too ready acceptance of any new system or method 
that comes along, especially if it is astutely advertised. It is 
perhaps one of our national weaknesses to swallow a nostrum 
too readily, whether it be a new patent medicine or a new method 
of teaching. What good reason have we for thinking that a 
teacher of 1931 is a more effective teacher than one of 1881, or 
for that matter of 2000 years ago? What is there in method, or 
in personal intelligence, that can give us any claim to be better 
teachers, better teachers, than were our forefathers? When 
a new method is announced, especially if it be announced with 
trumpets and shawms, write to the nearest Professor of 
Education, and more likely than not he will be able to give 
you the exact position of the old tomb which has been recently 
ransacked. 

A method which is outlined in a lecture or in a book is 
only the shadow of its real self. A method is not a piece of 
statuary, finished ~and unalterable, but is an_^er^chajniging 
thing, varying with the genius of the particular teacher who 
handles it. It is doubtful wisdom to try to draw a sharp 
antithesis between good methods and bad: the relative values 
of abstractions are invariably difficult to assess. The true 
antithesis is between eff^tiy^jfli^^^ The 

method itself counts for something, but what counts for very 
much more is the life that the craftsman whenjictually at work 
breathes into it. 

The regular working of mathematical exercises is essential, 



PREFACE xiii 

for the sake not only of the examination day that looms ahead 
but also for illuminating ideas and impressing these on the 
mind. Nevertheless, the working of exercises tends to dominate 
ourwork far too much, and to consume tiifte that might far 
more profitably be devoted both to the tilling of now neglected 
ground of great interest and to the serious teaching of the 
most fundamental of all mathematical notions, namely, those of 
number, function, duality, continuity, homogeneity, periodicity, 
limits, and so forth. If boys leave school without a clear grasp 
of such fundamental notions, can we claim that their mathe- 
matical training has been more than a thing of shreds and 
patches? I plead for a more adequate treatment of these things. 
The terms " Forms " and " Sets " I have used in accord- 
ance with their current meaning. The average age of each 
of the various sections of Form II, III, IV, and V is considered 
to be 12+, 13+, 14+, and 15+, respectively, the units figure 
of the age representing the Form: this is sufficiently accurate 
for all practical purposes. The age range of Form VI is taken 
to be 16+ to 18+. " Sets " represent the redistributed mathe- 
matical groups within any particular Form; for instance, 
100 boys in the different sections of Form IV might be re- 
distributed into 4 Sets, a, /?, y, and S. Admittedly it is in the 
lower Sets where skilled craftsmanship is most necessary. 

On reading through the manuscript I find that I have 
sometimes addressed the teacher, sometimes the boy, rather 
colloquially and without much discrimination. I crave the 
indulgence of my readers accordingly. 

Ait teachers of mathematics should belong to the Mathe- 
matical Association. They will then be able to fraternize 



xiv PREFACE 

periodically with the best-known and most successful of their 
fellow- workers. The Mathematical Gazette will provide them 
with a constant succession of lucidly written practical articles, 
of hints and tips, written by teachers known, by reputation if 
not personally, to everybody really interested in mathematical 
education; also with authoritative reviews of new mathematical 
books. Members may borrow books from the Association's 
Library, and the help and advice of specialists are always to 
be had for the asking. 

In writing the book my own views on numerous points 
have been checked by constant reference to Professor Sir 
Percy Nunn's Teaching of Algebra and its two companion 
volumes of Exercises, books I have not hesitated to consult 
and to quote from, in several chapters. I am old enough to 
remember the great reputation Professor Nunn enjoyed as a 
gifted teacher of mathematics when he was an assistant master 
thirty years ago. The methods he advocates are methods which 
have been amply tested and found to be sound and practical. 
His book deals with algebra in the broadest sense and gives 
advice on the teaching of trigonometry, the calculus, and 
numerous other subjects. 

F. W. W. 



CONTENTS 



CHAP, 
I. 

II. 

III. 

IV. 

V. 

VI. 

VII. 

VIII. 

IX. 

X. 

XL 

XII. 

XIII. 

XIV. 

XV. 

XVI. 

XVII. 

XVIII. 

XIX. 

XX. 

XXI. 

XXII. 

XXIII. 

XXIV. 

XXV. 

XXVI. 



TEACHERS AND METHODS ------ 

WHICH METHOD: THIS OR THAT? - 

" SUGGESTIONS TO TEACHERS " 

ARITHMETIC: THE FIRST FOUR RULES - ... 
ARITHMETIC: MONEY ------- 

ARITHMETIC: WEIGHTS AND MEASURES - 
ARITHMETIC: FACTORS AND MULTIPLES - 

ARITHMETIC: SIGNS, SYMBOLS, BRACKETS. FIRST NOTIONS 

OF EQUATIONS 

ARITHMETIC: VULGAR FRACTIONS - - - - - 
ARITHMETIC: DECIMAL FRACTIONS - ... 
POWERS AND ROOTS. THE A B C OF LOG- 



RATIO AND PROPORTION - 
COMMERCIAL ARITHMETIC 



ARITHMETIC: 
ARITHMS 

ARITHMETIC: 
ARITHMETIC: 
MENSURATION -------- 

THE BEGINNINGS OP ALGEBRA 

ALGEBRA: EARLY LINKS WITH ARITHMETIC AND GEOMETRY 

GRAPHS - - 

ALGEBRAIC MANIPULATION ------ 

ALGEBRAIC EQUATIONS 

ELEMENTARY GEOMETRY ------ 

SOLID GEOMETRY ----... 

ORTHOGRAPHIC PROJECTION 

RADIAL PROJECTION ....... 

MORE ADVANCED GEOMETRY .... 

GEOMETRICAL RIDERS AND THEIR ANALYSIS - - - 
PLANE TRIGONOMETRY ...... 



Page 
1 

16 
20 

23 
42, 
47 
54 

62 
67 

78 

94 
100 
109 
118 
122 
132 
137 
177 
205 
225 
287 
293 
300 
308 
319 



XVI 



CONTENTS 



CHAP. Page 

XXVII. SPHERICAL TRIGONOMETRY - - - - - 381 

XXVIII. TOWARDS DE MOIVRE. IMAGINARIES - 387 

XXIX. TOWARDS THE CALCULUS - - - - - 401 

XXX. THE CALCULUS. SOME FUNDAMENTALS - - - 421 

XXXI. WAVE MOTION. HARMONIC ANALYSIS. TOWARDS 

FOURIER -------- 451 

XXXII. MECHANICS 480 

XXXIII. ASTRONOMY 497 

XXXIV. GEOMETRICAL OPTICS 506 

XXXV. MAP PROJECTION 517 

XXXVI. STATISTICS 638 

XXXVII. SIXTH FORM WORK 653 

XXXVIII. HARMONIC MOTION 558 

XXXIX. THE POLYHEDRA 672 

XL. MATHEMATICS IN BIOLOGY 584 

XLI. PROPORTION AND SYMMETRY IN ART - - - 688 

XLII. NUMBERS: THEIR UNEXPECTED RELATIONS - - 694 

XLIII. TIME AND THE CALENDAR 614 

XLIV. MATHEMATICAL RECREATIONS 615 

XLV. NON-EUCLIDEAN GEOMETRY - - - - -617 

XLVI. THE PHILOSOPHY OF MATHEMATICS - - - 623 

XLVII. NATIVE GENIUS AND TRAINED CAPACITY - 639 

XLVIII. THE GREAT MATHEMATICIANS OF HISTORY - - 643 

XLIX. MATHEMATICS FOR GIRLS 644 

L. THE SCHOOL MATHEMATICAL LIBRARY AND EQUIP- 
MENT 646 

APPENDICES 
I. A QUESTIONNAIRE FOR YOUNG MATHEMATICAL 

TEACHERS 663 

II. NOTE ON AXES NOTATION 657 

INDEX 659 



PLATE 



POLYHEDRAL MODELS 



Facing 
Page 

- 584 



CHAPTER I 

Teachers and Methods 

Mathematical Knowledge 

" That man is one of the finest mathematicians I have 
ever had on my Staff, but as a teacher he is no good at all." 

" Oh yes, he can teach all right. He can hold a class of 40 
boys in perfect order for an hour. The trouble is that his 
knowledge of mathematics is so superficial." 

These criticisms of Head Masters are not altogether un- 
common. A Head Master naturally looks for two things in 
members of his Staff: sound knowledge of a subject, and skill 
in teaching that subject. 

Suppose that a teacher has spent three or four years at 
the University, has obtained the coveted First in mathematics, 
and has then spent a year in the Training College Department 
of a University or of a University College. Can he then claim 
to be a competent mathematician and a skilful teacher? The 
answer is no. 

The knowledge of mathematics obtained in a four years' 
University course forms an admirable foundation on which to 
build, but how much mathematics can be learnt in so short a 
period as four years? At the end of that time it is a common 
thing for University students hardly to have touched the serious 
mathematics of physics, of chemistry, of engineering, of ma- 
chinery and structures, of aviation, of astronomy, of biology, 
of statistics, or to have mastered more than the barest elements 
of the philosophy of the subject. Assuming that it takes a year 
to acquire an elementary knowledge of each of the applied 

(E291) 1 2 



2 CRAFTSMANSHIP IN MATHEMATICS 

subjects just named, the newly-fledged graduate has still in 
front of him a long spell of hard work before he can claim to 
be a mathematician in the broader sense of the term. As for 
the philosophy of mathematics, he might still be a babe in 
the subject after five years' hard reading. Mathematics touches 
life at so many points that the all-round training of a mathe- 
matician is a very long business. If by the age of 35 a mathe- 
matician has acquired a fair general knowledge of his subject, 
he has done well. Consider the mathematics of physics alone: 
what a vast field! The field is, of course, ploughed up and 
sown by the teachers of physics, with the result that there 
is always a rich harvest for the mathematical staff to reap. 
Is that reaping always done? 

Skill in Teaching: Training 

Then as to teaching. How can a mathematical Training 
College student claim to be an efficient teacher at the end 
of his training year? Skill in any art can be acquired only by 
much practice, and the art of teaching is_a particularly diffi- 
cult art. Although mathematics is one of th^BaQioot,- perhaps 
the easiest, of all subjects to teach, it is a rare thing for a 
mathematical teacher to be able to feel at all satisfied with 
his professional skill before the age of .30. He is lucky if 
other people adjudge him efficient before the age of 35. Every 
mathematical subject is full of teaching problems. Every 
one of these problems can be solved in a variety of ways. 
Every one of these ways is worth testing. And all this takes a 
long time. As for lucidity of presentation a prime necessity 
in all mathematical teaching that is in itself an art which 
half a life-time does not seem long enough to perfect. 

I have often been asked by mathematical: teachers who have 
not been formally trained what good they would have done 
by going to a Training College. They are inclined to argue 
that very few of the Training Colleges have on their Staffs 
a front-rank mathematician, and that therefore such Training 
Colleges are not in a position to deal with the subject effec- 



TEACHERS AND METHODS 3 

tively; that even if the Training Colleges are able to impress 
into their service members of the mathematical staff of the 
local University, little real help is obtainable, for " although 
a University Professor can teach me mathematics, he cannot 
teach me how to teach boys mathematics ". The general con- 
tention is not without its points, but the strictures certainly 
do not apply to all Training Colleges, as trained mathematical 
teachers who have been through the hands of, say, Professor 
Sir Percy Nunn are the first to admit. 

The further criticism that the time spent on Psychology at 
the Training Colleges is " absolutely wasted ", since " it has 
no practical value in the solution of actual classroom prob- 
lems ", is, perhaps, rather more justified. It is possibly true 
that the almost useless introspective psychology of half a 
century ago still hangs about some of the Training Colleges; 
I do not know. But young teachers should make themselves 
acquainted with the valuable experimental work which is now 
being done by psychologists all over the world. These ex- 
periments are often based on masses of actual data derived 
from the classroom. It is true that the definite results obtained 
so far are rather patchy; a complete body of psychological 
doctrine has yet to be built up into something that may claim 
to be " science ". But no teacher can afford to ignore the 
work that has been done and is being done, if only because 
such a large part of it has a very close bearing on present-day 
school practice. 

One broad distinction between the outlook of a teachei 
who has been through a Training College and that of one 
who has not is that the trained teacher has usually had knocked 
QUL.Qf_hini_the jprejudice which he (very naturally) felt for 
his own special subject, whether mathematics, classics, or 
what not. He has learnt that in the Common Room he will 
become a^ member of a community regresenting^alj the subjects 
of the curriculum, and that " all these subjects, not : Jiis own 
subject^alone, answer to dee-seated neejs_ofjhe human 
spirit, all of tftern es^ in the great stream of 

movement called^ civilization ". The Training College does 



4 CRAFTSMANSHIP IN MATHEMATICS 

its best, of course, to turn out competent craftsmen, but it 
does much more. It leads its students to understand the 
real meaning of education and something of its significance 
in relation to the many-sided business of life. It shows them 
how much wider education is than mere teaching. Moreover, 
the students are day by day in contact with men who have 
reflected deeply upon both education in the broader sense 
and teaching in the technical sense. 

In a good Training College, the embryo mathematical 
teacher is taught not only how to convey to his pupils a know- 
ledge of arithmetic, algebra, geometry, and the rest, but also 
how to make himself, in the greatest measure possible, an 
active intellectual adventurer in the realms of number and 
space, how to follow up the labours of the great masters of 
mathematical thought, and how to catch something of their 
spirit and outlook. He is encouraged to question accepted 
mathematical values, and to inquire, in a critical spirit, what 
parts of the traditional curriculum are really vital and what 
parts have only a conventional value. He is made aware that 
many of the textbooks contain a considerable amount of useless 
lumber, and he is taught: how to discriminate between methods 
that are sound and methods that are otherwise. 

Then again, in a good Training College the student is 
able to^obtain expert advice on every kind of difficulty he may 
meet with in his teaching practice. At what stage, for example, 
should " intuition " work in geometry give way to rigorous 
proof? How can the^best approach be made in the teaching 
of ratio and proportion? "How is the theory of parallels to 
be treated? The Training College may have to tell him frankly 
that in his present state of pupilage he is probably not yet 
fitted to deal with the theory of parallels except in an empirical 
way, since the inherent difficulties of the theory can only be 
grappled with after a prolonged and careful study of the re- 
searches of modern geometry; only then will he be in a position 
to disentangle logic from intuition, and so be able to devise a 
treatment suitable for Sixth Form boys. Briefly, the Training 
College will point out quicksands, as well as firm rock, for in 



TEACHERS AND METHODS 6 

mathematical teaching quicksands abound. A teacher who is 
not trained will often not only get into the quicksands but take 
his boys with him. 

It may be said that the great majority of Secondary School 
teachers, especially teachers in the Conference * schools, are 
not trained. That is true, and the result has often been that 
such teachers have bought their first few years* experience at 
the ^xpense of their boys. But not always. If a young man 
straight down from the University goes as a Student Teacher, 
that is as an observer and learner, in a large school where 
mathematics is known to be well taught, and if the Head of 
the mathematical department has sufficient leisure to guide 
him in his reading and to act as his professional friend, tjhtf 
year's training may not be inferior to that at a Training College, 
Some authorities urge that it may be superior, inasmuch as 
the Student Teacher spends the year with real practitioners 
rather than with theoreticians. I do not attempt to decide 
this question. 

Still another alternative a very common one is for the 
embryo teacher, just down from the University, to join the 
Staff of a big school, to admit freely that he is a neophyte, and 
to beg for all the help and criticism he can obtain from his 
mathematical colleagues. Let him invite his seniors to come 
to hear him teach, and to criticize him, and let him beg the 
privilege of being present occasionally at their lessons. Let 
him seek their advice as to a suitable course of reading. But 
let him not think that he is a teacher sent straight from heaven 
to rectify the views and methods of the old fossils of 25 and 
upwards he may find in the Common Room: that way lies 
trouble. 

One thing that the untrained beginner should never do 
is to join the Staff of a small school where he has to undertake 
the respcfnsfbffify of tR^whele of the mathematical teaching. 
In the first place it is not fair to the school; the boys are sent 
there to be taught. In the second place it is not fair to himself, 

*The term "Public" school is now ambiguous, and is better not used. All 
grades of schools are either "public" or "private". 



6 CRAFTSMANSHIP IN MATHEMATICS 

for how is he to learn his job? Why not open an office, and 
set up as a Consulting Civil Engineer instead? He is just as 
competent to do the one thing as the other. 

In the old days, all would-be craftsmen joiners, brick- 
layers, mechanics, and others served a seven years' apprentice- 
ship, and they really learned their business; they became skilled 
craftsmen. The system is dead, their work being done mostly 
by machinery. But we cannot teach by machinery yet; and 
skilled craftsmanship in teaching can be acquired only by a 
great deal of practice. 

Conventional Practice 

How is the value of a lesson in mathematics to be assessed? 
I do not refer to the ordinary things in which every teacher 
with a year's experience ought to be reaching efficiency 
class-management, discipline, use of the blackboard, expertness 
in questioning and in dealing with answers, and so forth 
but to the lesson as a mathematical lesson. The commonest 
fault of the young mathematical teacher is that he talks too 
much; he lectures, and, if he is teaching the Sixth Form, he 
often uses his University notes. It takes some young teachers 
a long time to learn the great lesson that the Jthing that matters 
jft.QSLJg. not what Jthey give out but what the boys take in; 
that their work is teaching, not preaching.""""" 

Another common fault of young mathematical teachers, 
and not all experienced mathematical teachers are guiltless of 
it, is the adoption of a particular method because it is mathe- 
matically neat, the sort of method that appeals to a mathema- 
tician as a mathematician, not the method that is the most 
suitable for demonstrating a particular principle or teaching 
a particular rule so that the child can understand it._ If a 
mathematical teacher thinks that a mathematician is listening 
to him, he is more often than not keener to reveal his know- 
ledge of mathematics than to exhibit his teaching power 
But the observer's object is usually to discover what the boys 
are learning, and to assess the value of the teaching; and the 



TEACHERS AND METHODS 7 

very u neatness " of the method adopted is quite likely to be 
the cause of the boys learning next to nothing. To be effective, 
a method must be simple and be clear to the boys. Mathe- 
matical rigour may thus have to Be""sacrlftced, though the 
rigour then sacrificed will come later. 

Again, young mathematical teachers are ajrt to be hide- 
bound b^coiwe^ . Generation after generation of boys 
are told, for example, that, in an expression like m + n X p, 
the multiplication sign should take precedence over the addition 
sign. Why should it? Doubtless the original suggestion is 
hidden away in some old textbook, but it has been consistently 
adopted by modern writers as if it were something sacrosanct. 
Surely if the signs are not to be taken in their natural or3er 
from left to right, it is the business of the person who frames 
the question to insert the necessary brackets, and not leave 
the wretched little learner to do it. 

Let the Head of a mathematical department in a big school 
remember that the place of honour for himself or for any 
exceptionally gifted member of his Staff isjn the lower Forms. 
The beginners' geometry is, more than any other mathematical 
work, in need of skilful teaching. The hackneyed stuff usually 
doneT by the Sixth Form specialists can quite well be taken 
by a youngster just down from the University. He may not 
be able to teach, but he is mathematically fresh, and, if the 
specialists in the Sixth have been previously well trained, 
they can usually take in the new mathematics even if it is 
rather clumsily presented to them. 

The textbooks mentioned in the course of this volume 
are intende^Tm the main, foLJhose teachers to read, who are 
technically untrained. The object is not to recommend this 
'Book "or that took for adoption in schools: that is not part oi 
our purpose at all. The object is to suggest a book, written 
by some skilful teacher, for the novice to read right through, 
critically. He should ask himself why the writer has approached 
and developed the subject in that particular way. He should 
then read a second book, then a third, and so on, noting the 
different ways of approach and of development, and the different 



8 CRAFTSMANSHIP IN MATHEMATICS 

ways in which different teachers do things. Then he should 
settle down and evolve methods for himself. He should not, 
unless in exceptional circumstances, copy another teacher's 
method. Let his methods be part of himself, things of his 
owiT elation, things for which he has an affection because 
they are his own children. 

Let him realize that methods of teaching mathematics, as 
of teaching other subjects, are largely conventional. What is a 
" best " method, and how is it to be determined? Is it a specially 
" neat " method, invented by some clever mathematician? If 
so, is it a simple method? Is it productive of accuracy? Here, 
psychology teaches us a little, though not yet very much, 
and to say that one method is " better " than another is, more 
often than not, merely to express a personal preference. The 
teacher should always ask himself, which method works out 
best in practice! Let every teacher make up his own mind, 
and not be led away either by the textbooks or by the critics, 
though the textbooks will always help, and the critics, if 
competent, are worth listening to. But, however good the 
books and however competent the critics, let him take their 
help and advice critically. 

Psychology kas k?lE<Lus great deal over certain points in 
the teaching^ oT arithmetic. Experiments have been directed 
marnly'to discovering which of possible alternative methods 
is productive of greatest accuracy amongst children. If such 
experiments are sufficiently numerous and varied, and if the 
results of the tests are fairly uniform, we may feel it advisable 
to consider a particular method favourably. But people who 
experiment in this way must set out with an entirely unpre- 
judiced mind. Results that are not arrived at objectively 
carry no weight. 

Whether psychology has yet succeeded in devising con- 
vincing testsjpf personal matb^atkal^ ability, I am un- 
certain. The validity of some of the criteria used has been 
seriously questioned by recognized authorities. Thejrelatipns 
between mathematical ability and general " intelligeace " 
have certainly not been clearly determined. We have probably 



TEACHERS AND METHODS 9 

all met highly intelligent men with keen logical powers who 
were no good at all at mathematics, and have known brilliant 
mathematicians whose lack of general intelligence in non- 
mathematical affairs was amazing. We do not yet really 
know if mathematical ability can be trained, or whether it is, 
so to speak, a fixed quantity at birth. The deductions we can 
legitimately draw from mere examination successes are by 
no means certain; even poor mathematicians may become 
adepts in the use of crammers' dodges. 

The "Dalton Plan " 

The question is sometimes asked, what is the Dalton 
plan of teaching, and can it be made to apply to mathematics? 

The plan originated in America ia 1920 and has since 
been introduced into a certain number of English schools. 
" The aim is to provide for the differences encountered in 
individual pupils." Class teaching as such is abolished, and 
gives way to organized private study, ijn^wWch^jhe_Uil, 
not the teacher, becomes the principal and responsible agent. 
Instead oFT course of lessons prescribed according to a 
time-table, an " assignment " of work, to last for a month, 
is prepared by the teacher. The whole " plan " hinges on 
these assignments. The month's task is divided into four 
weekly allotments, which are further subdivided jnto daily 
units. Instead of working to a time-table, the pupil is free 
to work at whatever subject he pleases. The rooms are no 
longer " classrooms ", but subject rooms, each being in 
charge of a specialist teacher and being provided with the 
necessary books and material. The pupils move freely about 
from room to room. The instructors are consulted at any 
time by any pupil; it is their duty to advise and help when- 
ever asked to do so. Conferences and collective discussions 
are, Jipwever, arranged at specified Tipurs. 
""""' There is a certain amount of acceptable opinion in favour 
of the plan as regards subjects like English and History, but 
as regards Mathematics, Science, and Modern Languages, 



10 CRAFTSMANSHIP IN MATHEMATICS 

the balance of opinion is undoubtedly against it. For one 
thing, theTmajority of mathematical textbooks are unsuitable; 
they do not demonstrate and elucidate principles simply 
enough for average pupils to understand, with the conse- 
quence that, in some schools which are working on Dalton 
lines, formal lessons on new principles precede the work by 
assignments, which, for mathematics, are not much more 
than a few general directions, and exercises to be worked. 

In short, the plan does not at present seem to be favoured 
very much by the majority of teachers. A teacher who adopts 
the plan, no matter what his subject, must be prepared for 
greatly increased personal labour; if his subject is mathe- 
matics, he must be prepared for some measure of disappoint- 
ment too. On the other hand, the able mathematical boy, if 
given a free hand, with just occasional help when difficulties 
are serious, seems to run away quickly from all the others. 
The plan seems to pay with Sixth Form specialists, who have 
been^weil trained up to the Fifth. Such boys, if provided 
with good textbooks, can, with very little formal teaching or 
other help, make remarkably rapid progress. 

The one general conclusion that seems to emerge from 
Dalton experiments is that pupils would do better if left to 
wrestle more for themselves, anckthat in the past we have all 
tended to teach too much. Although thejplan as ajglan is, in 
the estimation of not a few good judges, rather_top revolu- 
tionary for gejrier_aJLadpptipn, it must, on the other hand, be 
admitted that a clever teacher who loves teaching for its own 
sake may be something of a danger; hejrnay do too much 
of the thinking, and leave the boys too little to do for them- 

- 



No boy can become a successful mathematician unless 
he rights hard battles on his own behalf. 



TEACHERS AND METHODS 11 



Some General Principles 

The last statement does not mean that mathematical 
teaching is not necessary. For all pupils save perhaps the 
very best, it is fundamentally necessary, and above all things 
the teaching must be clear. Strive day by day to make the 
expression of your meaning ever clearer. Choose your words 
carefully and use them consistently. Never mind the correct 
formal definitions of difficult terms. Use a term over and 
over again always in exactly the same sense but associated 
in different ways with different examples, until its exact 
significance imposes itself on the pupil's mind. It is merely 
a question of the child continuing to learn new words much 
in the same way as he learnt the stock of common words 
which are already in his possession. His mother did not 
define for him as a baby such words as milk, mamma, toes, 
pussie, sleep, naughty, yet he learnt to understand their 
meaning almost before he could walk. 

In short, dojiot worry beginners with formal definitions, 
or abstractions of any other sort. Of course, almost from 
the first, the boy makes crude use of ajl sorts Q.f crudely 
acquired abstract terms, for in his enumeration work and in 
his early quantitative measurements, which he has always 
associated with concrete objects, intuition and guesses have 
played a large part. But the mathematical ideas and pro- 
cesses which he uses for solving different practical problems 
gradually become clearer, and he begins to see interrelations 
between principle and principle, and to distinguish those 
which are mutually connected from those which are inde- 
pendent. As the subject proceeds, it tends to become more 
abstract; experience grows; and the teacher has to choose 
his own time for stepping in and exacting greater and greater 
logical rigour. Below the Sixth Form, mathematics is essen- 
tially a practical ^^ instrument, not a subject for philosophic 
speculation. Never press forward formal abstract considera- 
tions until ' ~~ "*- 



12 CRAFTSMANSHIP IN MATHEMATICS 

What is the use of discussing even with Fifth Form pupils 
the rival merits of Euclid's parallel postulate and Playf air's 
alternative version? For all lower and middle forms, some 
such statement as, " lines which intersect have different 
directions; lines which have the same direction do not meet 
but are parallel ", is good enough, and it need not be sub- 
jected to criticism until the Sixth. Then, criticism is desirable. 

The organization of mathematical work in a large school 
is a simple matter; between the Junior Forms and the Sixth 
there may be 4 blocks of 3 or 4 Sets each. When the Set 
system prevails, gradation is easy. Let the work of the top 
Set of a block be much sterner and more exacting than in 
the bottom Set, and do not attempt to include in the work of 
the bottom Set all the subjects, or even all the topics of a 
particular subject, that are allotted to the better Sets. For 
instance, all top Sets will learn logarithms. But bottom Sets? 
Why should they? What difficult calculations will they have 
to engage in that logarithms will really help? None in school, 
and none after leaving school. Why then should such dull 
boys be made to waste their time by poring over the pages 
of a numerical lexicon and then getting their sums wrong 
instead of right? It is unutterably silly. It is sometimes done 
because teachers have not the courage to say what they really 
think. 

The timid teacher may be inclined to argue, " but how are 
we to provide for the boy who during the year happens to be 
promoted a Set?" That is certainly a real problem of school 
organization and must be faced. But the needs of the occa- 
sional boy must not be catered for at the expense of a whole 
class. And, after all, there will be much the same minimum 
of work for the various Sets within a block, and inter-block 
promotions after the first year or two will be rare. 

Again, suppose that somebody comes along and asks if 
you teach, say, Vectors. If you do not, you probably have 
a good reason for it, perhaps because Lord Kelvin himself 
poured scorn on them. In that case do not hesitate to say so. 
Holdfast to your faith. But re-examine the grounds of your 



TEACHERS AND METHODS 13 

faith from time to time. It may be that you will find new 
arguments in favour of vectors, arguments which will induce 
you to revise your opinions. And so with scores of other 
things. Keep an eye on your defences, but remain captain 
of your own quarter-deck. 

Mathematical Reasoning 

All mathematical teachers should reflect carefully on the 
nature of mathematical reasoning, and should see that their 
pupils are made more and more conscious of what constitutes 
mathematical rigour. Mathematical reasoning is not, as com- 
monly supposed, deductive reasoning; it is based upon an 
initial analysis jrf jthjy^en. and, being analytical, is,jn essence 
' ~* " ' e. The threads of the web once disentangled, synthesis 



begins, and the solution of the problem is set out in deductive 
4yqgs. We arrange our^ arguments ded^ 

y easily^jFollow. up the chain to .oiir^ .final 



conclusion. If this mere setting ouTVere the whole story, 
how simple it would be! Consider this syllogism, in form 
typically Euclidean and deductive: 

Major premiss: All professional mathematicians are 
muddle-headed. 

Minor premiss: The writer of this book is a professional 
mathematician. 

Conclusion: Therefore the writer of this book is muddle- 
headed. 

Now the conclusion is quite possibly true, and it is cer- 
tainly the correct conclusion to be drawn from the two pre- 
misses. But both the major and the minor premisses are 
false (the writer of this book is not a professional mathe- 
matician: heaven forbid! he is only a teacher), and therefore 
the conclusion, even if materially true, is logically absurd. 
In fact the main source of fallacious reasoning almost always 
lies in false premisses. The truth of the conclusion cannot 
be more true than the truth of the premisses, and^ a scrutiny 
and a rigorous analysis of tfi^ap is tWefore always necessary. 



H CRAFTSMANSHIP IN MATHEMATICS 

At bottom, all reasoning is much of the same kind, and 
it usually turns on the truth or falsehood of the premisses. 
Clear thinking is thus indispensable: probabilities have to be 
weighed, irrelevant details discarded, the general rules accord- 
ing to which events occur have to be divined, hypotheses have 
to be tested; the general rules once established, the derivation 
of particular instances from them is a simple matter. 

But in elementary mathematics fer beginners, the_prQiipn 
of concrete particular instances comes first in importance. 
In the handling of his subject in the classroom, the mathe- 
matical teacher cannot be too concrete. As the boys advance 
from Form to Form, they will gradually begin to understand, 
and in the Sixth to realize fully, that the solving of every 
mathematical problem consists first of disentangling, then of 
setting out and classifying, then of tracing similarities and 
finding possible connecting links, then of linking up and 
generalizing; in other words, o analysis followed by syn- 
thesis. Although without generality there is no reasoning, 
without concreteness there is neither importance nor signifi- 
cance. But in schools logical rigour is a thing of exceedingly 
slow growth. We shall return to the question of mathematical 
reasoning in a later chapter. 

More often than not, present-clay writers of standard text- 
books in mathematics strain after both ultra-precision of 
statement and the utmost rigidity of proof. But any attempt 
in schools to be perfectly exact all at once, to include in every 
statement all the saving clauses and limitations that can be 
imagined, inevitably ends in failure. This is where the be- 
ginner, untrained, just down from the University, so often 
blunders. He is inclined to argue that, unless his classroom 
logic is as unassailable as that of his University Professor, 
his work will be open to serious criticism. The work of even 
Sixth Form specialists cannot be placed on an unimpeachable 
logical basis. The_ degree of riggur_ r tiiat,caa.,h.jixacted at 
apyjrtage must necessarily ^depend on the degree of intellectual 
development of^ilie pupil. A school can never become a 
place for mathematicaTasceticism, 



TEACHERS AND METHODS 15 

But, as boys get older, they should be encouraged to read 
their own textbooks " up and down, backwards and forwards ". 
In their study, let us say, of the calculus, let them first obtain 
an insight into general elementary processes, and then pro- 
ceed at once to simple applications. Ample practice in 
differentiation and integration is, of course, necessary, but 
the study of geometrical and dynamical applications must not 
be unduly delayed. It is these that will excite interest, and 
will help greatly to produce an appreciation of fundamental 
principles. But again and again go back to a more critical 
examination of those principles. The applications will have 
taught the boys a great deal of the inner meaning of the pro- 
cesses, and the more abstract discussions will then be made 
much easier by the fact that the learner has acquired a fair 
stock of more or less concrete ideas. 

The Fostering of Mathematical Interest 

The general standard of mathematical attainments in 
Sixth Forms is now reasonably satisfactory, and entrants at 
the newer universities are beginning work of much the same 
grade as entrants at Cambridge. But though Sixth Form 
specialists are doing solid work (of a very restricted type, it 
must be added), the amount of mathematical work being done 
by all the other pupils who have obtained the School Certifi- 
cate is, as a rule, slight, too slight and much toojacademic for 
the fostering,,, of a Jife-Jong interest in the subject. Let the 
younger race of teachers wake up to this important fact, and 
help to put things right. We shall refer to this point again. 

Books to consult: 

1. Didaktik des mathematischen Unterrichts, Alois Hofler. 

2. A Study of Mathematical Education, Benchara Branford. 



16 CRAFTSMANSHIP IN MATHEMATICS 

CHAPTER II 

Which Method: This or That? 

Old and New. Rational and Rule-of-thumb. 

An intelligent woman, who is known to have done a fail 
amount of mathematics in the days of her youth, recently 
received a bill for 8s. 7rf., representing the cost of 7 Ib. 6 oz. 
of lamb. She was " sure " that the ounces and farthings had 
been included merely for the purpose of cheating her, and 
she telephoned to the butcher to know the price of the meat 
per Ib. She was quite unable to calculate the amount for her- 
self (Is. 2d.). 

A well-known Inspector of the nineties dictated this sum 
to a class of 11-year-olds: " Take one million ten thousand 
and one from ten millions one thousand one hundred ". As 
might be expected, hardly any children had the sum right. 
The Inspector looking grieved, the Teacher gently asked him 
if he would himself work the sum on the blackboard. Very 
unwisely the Inspector tried to do so, and made a hopeless 
mess of it to the delight of the boys. 

The first story illustrates ojiejDf the commonest faults of 
scBool mathematics: teachers are apt to 'push on into more 
advanced work before foundations have been well and truly 
laid. The second story shows that a^non-mathematician 
should not be allowed to criticize mathematical teacher^. To 
the non-specialist, mathematics is full of pitfalls, and it may 
be hoped that the time will come when every teacher of the 
subject will be a trained mathematician, even if he has to 
teach nothing but elerrientary arithmetic. 

Not the least important question for a teacher of elemen- 
tary arithmetic to consider is the method ...of ._aBPtflacb , tQ . a 
new rule. Should that rule be given to the child dogmatically, 
given as a rule, to be followed by the working of examples 



WHICH METHOD: THIS OR THAT? 17 

until it is thoroughly assimilated? or should the rule be 
" explained ", approached " intelligently ", and be thoroughly 
" understood ", before it is applied to examples? In other 
words, is it immoral or is it legitimate to provide a child with 
a working tool before the nature of the tool is explained? 

To put it another way: suppose that we teach a rule 
" intelligently ", and the children get 50 per cent of their 
sums right; or suppose that we teach by rule of thumb and the 
children get 80 per cent of their sums right. Which plan 
should we adopt? 

Should we give credit merely for " getting sums right "? 
or should we forgive mere slips if the working shows some 
grasp of the process? 

Again: suppose we find that some of the newer and 
popular methods, methods that have superseded those in 
common use forty or fifty years ago, are less productive of 
speed or accuracy or both, are we, or are we not, justified in 
feeling a little suspicious of the newer methods? 

Some of these questions have been answered for us by the 
psychologists, who in recent years have adopted various 
devices for testing the comparative merits of the methods 
we use in teaching arithmetic. The old school of psycholo- 
gists trusted t&cKrnuch to intuition, and their views were 
doctrinaire. ^Present-day psychologists, on the other hand, 
are devoting themselves to experiment, to_the garnering of 
facts, to making careful deductions from those facts. For 
instance, some of them have**ar ranged with schools for tens 
of thousands of simple sums, of varying types, to be worked 
by different methods. From such large numbers of results 
legitimate deductions may be drawn, especially when different 
psychologists arrive at similar conclusions from different sets 
of examples. It is on such evidence as this that different 
methods have been compared and some sort of priority 
determined. No thoughtful mathematical teacher would now 
pronounce dogmatically in favour of. his own method of 
doi^ pithing, even if he has used it : all his life. 'pHe would 
suEJect it, ancl other methods as well, to prolonged tests 

(B291) 



18 CRAFTSMANSHIP IN MATHEMATICS 

selecting different groups or classes of children all " new " 
to the principle to be taught; and he would compare the re- 
sults in different ways, for instance for intelligence, for 
accuracy, and for speed; and he would make sure that the 
general conditions of the tests, for instance the time of day 
when they were given, were equalized as far as possible. It 
is in such matters that psychologists are helping us greatly. 

Perhaps the first essential of all is accuracy, especially 
Accuracy in all kinds of computation. What would be the 
use oT"a bank clerk who^ made mistakes in running up a 
column of figures? A tradesman inaccurate in his calculations 
might jsoon ^find hunselt a-bankrupt . Indeed , accuracy jranjjs 
as a ^cardinal viytiie^ it Jgja main factor of morality. A boy 
who gets a^smnjw rongjshQ.uld~ be made to get it right. Never 
accept a wrong answer. This does not mean that credit 
shbuldTTot be given for intelligence: anything but that. For 
instance, a boy may be given a stiff problem and get it wrong. 
But that problem may include half a dozen little independent 
sums, each of them to be thought out before it can be actually 
worked; five of them may be right and one wrong. In such 
a complex operation, a margin of error may be legitimately 
allowed for. 

If we think of our own personal operations in arithmetic, 
those we are engaged in day by day, we must admit that most 
of our working is by rule of thumb; the actual rationale of a 
process does not enter our heads. We have become almost 
mathematical automata. Yet, if called upon to do so, we 
couIcCof course, explain the rationale readily enough. But 
the average boy, the average boy, however intelligently he 
may have been taught, not^only works by rule of thumb but 
could not for the life pf jiij&.give~an adequate explanation pf 
tEe^ocess. This is admittedly brutal fact. Test any average 
class of 30 boys, twelve months after they have been taught 
a new rule, and it is highly improbable that more than 8 
or 10 will explain the mathematical operation adequately 
and intelligently. The experienced teacher never expects it. 

Nevertheless, no teacher worth his salt would ever dream 



WHICH METHOD: THIS OR THAT? 19 

q teaching a new rule j^ithout % approaching _Jt ratio.nally. 
He would do his best to justify every step of the process, 
illustrating and explaining as simply as possible. Perhaps 
4 or 5 of the boys in a class will see the whole thing clearly, 
and their eyes may sparkle with satisfaction. A few more, 
perhaps 8 or 10, will follow the argument pretty closely, 
though if asked to repeat it they will probably bungle pretty 
badly. But the rest? No. They want the rule, simply and 
crisply put, a rule they can follow, a rule they can trust and 
hold fast to. And no teacher need break his heart that the 
majority can do no more. No inspector, if he is a mathe- 
matician, ever expects more; he is too familiar with the 
mathematical limitations Jthat nature has imposed on the 
average boy; of British origfn^ ~ 

As a boy goes up the school and his intelligence is de- 
veloped, the fundamental processes of arithmetic may be 
made clearer to him. Any average boy of 13 or 14 may be made 
to understand the main facts of our decimal system of nota- 
tion, whereas at 7 or 8 he may have failed to grasp the real 
significance of even a three-figure number. Every teacher 
of mathematics should remember that he cannot clear the ground 
finally as he goes along\ he has to come back again and again. 

Do not worry young children with such terms as abstract 
and concrete. Nothing is gained by telling a child to add 8 
sheep to 9 sheep instead of 8 to 9. Actual arithmgtical pro- 
cessejy*e^ll abslrjcjt, j^djyhe^jnotion.^f , jcastingjsvjgry sum 
into problem fprm_baL^J?^QQm^ a_illy fetish. Present-day 
cEi!3ren are suffering from a smjeUjjf oranges and^apples. 
Of course when little children are beginning to count, to add, 
to subtract, &c., the use of real things is essential, and in this 
matter we may learn much from t he efficient 



teacfaer. Some of the very fcest arithmetic J^achJn&J . 

school^. It is a pleasure to watch chil- 



dren who are little more than toddlers getting a real insight 
into number and numeration. The worst teaching of arith- 
metic I have ever seen was in the lower forms of the old 
grammar schools of 40 years ago. In those days it was not 



20 CRAFTSMANSHIP IN MATHEMATICS 

an uncommon thing for the lowest forms to be placed in 
charge of an unqualified hack. Those were dark days indeed. 
" Practical " mathematicsjincludes manipulative work of 
some kind, actiiaf measuring as well as calculating, and the 
more of this irTtKT Seconds, Third^ and Fourths the better, 
especially if there isTa mathematical laboratory available. 
It is concrete mathematics, but do not give it that label. In 
fact, put the label into the waste-paper basket. As for the 
label abstract, burn it. 

Books to consult: 

1. The Approach to Teaching, Ward and Roscoe. 

2. The New Teaching y Adams. 

(These are not books specially directed to mathematics, but to 
teaching generally. They are books to be read by every teacher, 
for they are full of good things. Mr. Ward was for many 
years chief Inspector of Training Colleges; Mr. Roscoe is 
Secretary to the Teachers' Registration Council, and was 
formerly Lecturer on Education at the University of Birming- 
ham; Sir John Adams was formerly Professor of Education 
in the University of London.) 



CHAPTER III 

"Suggestions to Teachers" 

The Handbook of Suggestions to Teachers, 1928, issued by 
the Board of Education, contains useful hints " for the con- 
sideration of teachers and others concerned in the work of 
Public Elementary Schools ". The practical hand is revealed 
on every page, and there can be no doubt that the best teach- 
ing practice known in the country is embodied in it. The 
book deals specifically with the regui^ments jof Elementary 
Schools as they are likely to be developed during thePnext 
few years Infant ^Schools, Junior Schools, and Senior 



"SUGGESTIONS TO TEACHERS" 21 

Schools, including " Selective " Central Schools; but what 
is said about mathematics, especially arithmetic, is equally 
applicable to scKoofs oFaU types. 

The Board are of opinion that, by the age ofJLJU." a mini- 
mum course should at least include a thorough groundwork 
in notatigin, a knowledge of the %s^J^r_rjjle^^^ied[ to 
mpngy, and the ordinary English measures of length, area, 
capacity, weight, and jtime; an elementary acquaintance with 
vulgar and decimal fractions, together with simple notions 
of geom_etrigal form and some skill in practical measure- 
ments." By that age, " accuracy in simple operations should 
in great measure be automatic. It depends first on a ready 
Knowledge of tables,, and secondly upon concentration, but 
in the case of written work is greatly assisted by neatness of 
figuring and clear statement." 

(j^The Board contemplate that, in future, the arithmetic of 
all Senior Elementary Schools (where the age will extend 
from 11 to 14 or 15 or even 16) will be associated with men- 
Duration, scale ^drawing, geometry, gny>h, and (for boys) 
algebra, trigonometry, and practical mechanics.^ The course 
of mathematical work mapped out for such schools is par- 
ticularly suggestive and should be read by mathematical 
teachers in all schools. 

The Board seem also to contemplate for Senior Elemen- 
tary Schools some form of mathematical laboratory where 
practical work can be done. This "work is to be associated 
with the geometry, mensuration, surveying, mechanics, and 
manual instruction, and even for the lqyv^r_cla.s.es^.iiseful 
hints are given for practical. work in weights and nieasures. 
Much of the work which at one time constituted the pre- 
liminary course of practical physics might be included as 
well the use of the vernier and the micrometer screw gauge, 
the volumes of irregular solids by displacement, densities 
and specific gravities, U-tube work, and experimental veri- 
fications of such principles as those involved in the lever 
and pulleys, in the pendulum, and in Hooke's law. 

The time has gone by when arithmetic, even in Elemen- 



22 CRAFTSMANSHIP IN MATHEMATICS 

tary Schools, should be looked upon as a self-contained sub- 
ject. Although arithmetic is the subject dealing with numerical 
relations, it is geometry which deals with space relations, and 
tfuTtwo should be taught together. Algebra is just a useful 
mathematical instrument, full of 



devices for both arithmetic^and geometry. TTrigpnometry 
is the ^^ surveyor's subject ,jTTiseful appl^itojp_or^j^bra,afld 
geometry together. A graph is a geometrical picture, showing 
arithmetical amTalgebraic relations of some sort. The various 
subjects fuse tojjether as parts of a single puzzle^and quite 
young boys may be given a working insight into them all. 
Arithmetic alone is dry bread indeed, far too beggarly a 
mathematical fare even for a Junior School. 

From the first, keep the mathematical work _jn_close con- 
tact Ayith^thfe, problems, ^of practical life. Let matters reasoned 
about be matters with which the children are either already 
familiar or can be made to understand clearly. Po_ not, take 
the children for excursions into the clouds^ what is perfectly 
cleaFlo you may JBe^veiyJoggy to them.^ Therejs no^ inde- 
pen3ent " jaculty " oTreagpning, independent of the par- 
ticular facts and relations reasoned about, stored away ghost- 
like in the brain, to be called upon when wanted. Hence, 
always endeavour to ensure that the things which^ you call 
upon a boy to discuss are seen by him as in a polished mirror. 

of mathematical terms. 



If you take care always to use such terms in tHeir exact sense 
they need rarely be defined. Even very small children have 
to learn the terms add, subtract, sum, difference, remainder, 
whole, part, less, equal, equals, total, and older children m'ust 
acquire an exact knowledge of such terms as interest, discount, 
gross, net, balance, factor, prime, measure, multiple, and dozens 
of others. If you use them consistently, the children will 
soon learn to appreciate their exact significance. 

Let part of your stock-in-trade be price-lists of some of 
the big London stores, the Post Office Guide, Bradshaw, 
Whitaker's Almanack, and the like. Ask yourself what sort of 
mathematical knowledge the children are likely to require 



" SUGGESTIONS TO TEACHERS" 23 

inafter life, and, as far as you can r provide accordingly. But 
it_is not merely a^guestion of giving them practicaT tips: 
train them to think mathematically. Train them to_car^_Jor 
accuracy. Train them to appreciate some of the marvels of 
thelnirverse the very great and the very small. 

^ not ^ise .^Ijd-^sJuoned methods that have stood 



the test of time, and do not be too ready to adopt the new- 
fangled methods of some new prophet. Any new educational 
lubricant which is advertised to be a tremendous accelerator 
of the classroom machinery generally proves to be nasty 
clogging stuff, making life a burden for those who use it. 

Book to consult: 

Handbook of Suggestions for Teachers, H.M. Stationery Office. 



CHAPTER IV 

Arithmetic : The First Four Rules 

Numeration and Notation. Addition 

We have already mentioned that the laborious work of 
psychologists has taught us much about the pitfalls experienced 
by beginners when learning arithmetic. Few young teachers 
realize the number of separate difficulties felt by children in 
learning to do ordinary addition sums, even after the addition 
table to 9 -f- 9 is known. 

For instance, a child has to learn: 

(1) To keep his place in the column; 

(2) To keep in mind the result of each addition until 

the next number is added to it; and 

(3) To add to a number in his mind a new number he 

can see; 



24 CRAFTSMANSHIP IN MATHEMATICS 

(4) To ignore possible empty spaces in columns to 

the left; 

(5) To ignore noughts in any columns; 

(6) To write the figure signifying units rather than 

the total number of the column, specially learn- 
ing to write when the sum of the column is 
20, 30, &c.; 

(7) To carry. 

A teacher should analyse in this way every general arithmetical 
operation, and provide an adequate teaching of every separate 
difficulty. Unless at least the slower pupils are thus taught, 
they may break down in quite unsuspected places. Another 
important thing is the grading of difficulties. For instance, 
we now know that the average beginner finds the addition sum 

21 4 

43 easier than 21 

35 3 

35 

and the latter very much easier than 21+ 43 + 35. He seems 
to have more confidence in the completed columns ? and the 
vertical arrangement appeals more stronglyto his eye^ than 
does^T^horizonfal arrangement 

Here is a series of first subtraction sums, graded accord- 
ing to the difficulty experienced by beginners: 

888 
380 

Teach one thing at a time] see to it that this one thing does 
not conceal a number of separate difficulties; and let that one 
thing be taught thoroughly before the next is taken up. 

The bare elements of numeration and notation will have 
been taught in the Infant School or Kindergarten School, 
and on entry to the Junior School or Department the children 
will clearly apprehend the inner nature of a 3-figure number, 
that, for instance, 

357 = 300 + 50 + 7. 



ARITHMETIC: FIRST FOUR RULES 



25 



If that is thoroughly understood, but not otherwise, numeration 
and notation should give little further difficulty. 

Numbers of more than 6 figures will seldom be required 
in the Junior School or Department, and children soon learn 
to write down 6-figure numbers correctly. Let beginners 
have two 3-column ruled spaces, thus: 



Thousands. 


H. 


T. 


U. 


H. 


T. 


u. 


2 


4 


3 


5 


9 


6 



Tell them they have to fill up the spaces under " thousands " 
exactly as they fill up the old familiar 3-column space on the 
right. Dictate " 243 thousands ", and pause; the child 
writes 243 under " thousands ". Now go on: " listen to 
what comes after thousands; 596 ". The child soon learns 
to write down a dictated number of " thousands ", just as 
he would write down a dictated number of " books ". With 
a class of average children of 10 or 11 years of age, one lesson 
ought to be enough to enable them to write down even 
9-figure numbers accurately, if these are properly dictated, 
and if the children are first made to understand that after 
" millions " there must always be two complete groups each 
of 3 figures, the first of these groups representing thousands. 



Millions. 


Thousands. 


H. 


T. 


U. 


H. 


T. 


U. 


H. 


T. 


U. 





















The teacher dictates: " Write down 101 million 10 thousand 

and one." 
" How many millions?" " 101." " Write 101 under 

millions' 9 

" How many thousands?" " 10." " Write 10 under 
thousands" 



26 CRAFFSMANSHIP IN MATHEMATICS 

" What comes after thousands?" " 1." " Write 1 in the 
right-hand 3-column space." 



1 1 



1 



" Now fill up with noughts." 

Numbers without noughts should come first. Introduce the 
noughts gradually. Remember that they provide constant 
pitfalls for beginners. 

Dispense with the ruled columns as soon as possible, 
but let the successive triads always be separated by commas; 
14,702,116 (14 millions, 702 thousands, 116). 



Subtraction 

On entry to the Junior School the child will already have 
been taught that the subtraction sum 



is a shortened form of 



800 
100 



867 
142 
725 

60 

40 



700 + 20 + 5 



bundles of sticks or bags of counters being used to make the 
process clear. 

They will also have been taught to decompose the top 
line in such a sum as this, leaving the lower line alone: 



534 
386 



Thus: 



500 + 30 + 4 __ 400 + 120 + 14 

300 -j- 80 + 6 ~" 300 +80+6 

100+ 40+8 = 148 



ARITHMETIC: FIRST FOUR RULES 27 

I have often seen excellent results in such an instance, the 
small children handling their bundles or bags, untying them 
and regrouping, in a most business-like way. They really 
did seem to have grasped the essentials of the process. 

But can the method be regarded as the most suitable 
permanent possession for older children? Consider this sum: 

80,003 
47,167 



The necessary decomposition is a complicated matter for 
young children. They have to take 10,000 from the 80,000; 
leave 9000 of the 10,000 in the thousands column and carry 
1000 on; leave 900 of this 1000 in the hundreds column and 
carry 100 on; leave 90 of the 100 in the tens column and 
carry 10 on to the units column. And thus we have: 

80,000 + 3 70,000 + 9000 + 9004-90 + 13 

40,000 + 7000 + 100 + 60 + 7 40,000 + 7000 + 100 + 60+ 7 

30,000*+2000 + 800 + 30+ 6 = 32,836 

In practice this is what we see: 

7999 

** % q q'3 
4 7, 1 6 7 

"~ 



I confess that, judged by the number of sums right, the best 
results I have ever met with were in a school where this 
decomposition of the minuend was taught, although the 
teacher responsible was not only not a mathematician but was 
entirely ignorant of the principles underlying the plan she 
had adopted. She would give the children a sum like this: 

70,024 
30,578 

Before the children began actually to subtract, they had 
to examine each vertical column of figures, beginning with 
the units column. If the upper figure was smaller than the 



28 CRAFTSMANSHIP IN MATHEMATICS 

lower, they would borrow 1 from " next door ", prefix it 
to the unit figure in the top line, and show the borrowing 
by diminishing by 1 the figure they had borrowed from. 
Thus they wrote: 

l 

7 0, *{ 1 4 

3 0, 5 7 8 

Then they would examine the tens column. If, as before, 
they found the upper figure smaller than the lower, they 
would borrow from next door again if they could; if not, 
they would pass along to the place where borrowing was 
possible, in this case 7, change the 7 to 6, and prefix the 
borrowed 1 to the 1 in the tens column, always changing 
into 9's the O's they had passed over. They always " borrowed 
from next door when anybody was at home, putting a 9 on 
the door of every house they found empty ". Thus the sum 
was made to look like this: 

6 9, 9 11 

X q ^ 1^4 
3 0, 5 7 8 

Then the subtracting was begun, and, of course, it was all 
plain sailing: 8 from 14, 7 from 11, 5 from 9, from 9, 
3 from 6. 

Over and over again I tested that class, and not a child 
had a sum wrong. But the children had no idea of the " why " 
of the process; neither had the teacher. The accuracy was 
the result of a clear understanding of an exactly stated simple 
rule. The children followed the rule blindly. 

But this case does not typify my general experience, 
which is that the decomposition method, is not productive 
of anything like the accuracy obtained by the alternative 
method of equal additions. Although, therefore, I am drivgn 
to favour the equal additions method, this method does not 
seem quite so susceptible of simple concrete explanation 
for very young beginners. Still, such explanation is possible. 

First try to make the pupils understand that equal additions 



ARITHMETIC: FIRST FOUR RULES 29 

to the minuend and subtrahend * will not affect the difference. 
The ages of two children provide as good an illustration as 
anything: 

Jack is 7 and Jill is 10; their difference is 3. In 4 years* 
time, Jack will be 11 and Jill will be 14; their difference 
will still be 3. And so on. Their difference will always be 3. 

A first lesson on brackets will serve to reinforce the idea: 

10-7 =3 

(10 -f 4) - (7 -f 4) = 14 - 11 = 3 
(10 + 6) - (7 + 6) = 16 - 13 = 3 
Then 

73 73 + 5 = 78 73 + 10 = 83 

21 or, 21 + 5-26 or, 21 -f 10 = 5U 
52 52 52 

In this way, get the child to grasp the cardinal fact that in 
any subtraction sum we may, before subtracting, add any- 
thing we like to the top line if we add the same thing to the 
bottom line; the answer will always be the same. Another 
example: 

73 70 4- 3 70 -^13 

48 40 + 8 50+8 

Here we have added 10 to the top line, turning 3 into 13, 
and we have added 10 to the bottom line, turning 40 into 
50. (The double arrow usefully draws attention to the two 
additions.) Thus the answer to the altered sum will be the 
same as to the original sum. In this way it is easy to give a 
clear understanding of the so-called " borrowing " process. 

But the small child is not quite so happy when working 
by this method in the concrete, as he is with the decomposition 
method. When he is given the two extra bundles of 10, 
he does not always believe that the sum can be the same. 

* Do not use these terms with beginners; wait until the senior school. The /row 
line and take line will do, or the top line and bottom line, or the upper line and lower 
line. Adopt simple terms of some kind, and adhere to them until the children are 
better prepared to adopt a stricter nomenclature. 



30 CRAFTSMANSHIP IN MATHEMATICS 

However, some kindergarten teachers seem to have little 
trouble about it. 

In a sum like the following, the teaching jargon * we 
should use for beginners would probably be something like 
this: 

" Whenever wejyjve_l to thetop line, we musLalwapgive 

1 to the bottom line as well, but next door." 

9023 
3765 

" 5 from 3 we cannot; give 1 to the top line and so turn 
3 into 13; 5 from 13 is 8. Now give 1 to the bottom line, 
next door; 1 and 6 is 7." 

" 7 from 2 we cannot; give 1 to the top line, and so turn 

2 into 12; 7 from 12 is 5. Now give 1 to the bottom line, 
next door; 1 and 7 is 8." 

" 8 from we cannot; give 1 to the top line and so turn 
into 10; 8 from 10 is 2. Now give 1 to the bottom line, 
next door; 1 and 3 is 4. 

" 4 from 9 is 5." 

The words borrow and pay back tend to mislead the 
slower boys, since we borrow from one line and pay back 
to another. To them this seems unfair, especially when we 
say we borrow 10 and pay back only 1. 

Personally I prefer to give 1 to the top line and never 
talk about paying back, but compensate by giving 1 to the 
bottom line. But the 2 parts of each double transaction must 
be worked in association at once; this satisfies the children's 
sense of justice. 

In each of the first several lessons, ask what the giving 
of 1 really signifies. " When we turned 3 into 13, the 1 given 
was really 10; did we give the bottom line the same number? 

* The term jargon is rather suggestive of slang, but, of course, what I really mean 
is the simple homely language which we mathematical teachers all invent for teaching 
small boys, language which rather tends to offend the ear of the English purist. But 
that does not matter. The important thing is to express ourselves in words which 
convey an exact meaning to the children's minds, 



ARITHMETIC: FIRST FOUR RULES 31 

Yes, because the 6 which by adding 1 we turned into 7 
is in the tens column. 

" When we turned the 2 into 12, the 2 was really 20, 
and the 1 we gave to it was really 100; did we give the bottom 
line the same number? Yes, because the 7 which by adding 
1 we turned into 8 is in the hundreds column." 

And so on. A very nnsui era r>le proportion of the children 
will not at this stageunderstand the process at all. But do 
not worry about that. Come back to it in a Ygsy^Jimg* 

It will be weeks, even months, before the slower child 
will have had enough practice to do subtraction quickly and 
accurately, and it is best to adhere all the time to precisely 
the same form of teaching " jargon ". 

There remains the question, shall we teach the children 
(i) actually to subtract, or (ii), to add (complementary ad^- 
ditionj, or (iii), first to subtract from 10 and then add the 
difference to the figure in the top linel For instance, 

13 



Shall we say 5 from 13 is 8 i 
or, shall we say 5 and 8 is 13? ' 
or, shall we say 5 from 10 is 5 and 3 is 8? 

The last must be ruled out of court; it is productive of great 
inaccuracy amongst beginners, though later on it is useful in 
money subtraction. The jsecond^Elan is popular, but it has 
been proved to be less productive of accuracy than thejiffi; 
and it is Something ot' a sham, for thenumber to be added 
must be obtained by subtraction.* Hence the first method, 
honest subtraction, is strongly advocated, and that demands 
ample practice in both the subtraction and the addition 
tables. Thus the child learns: 



both 



5 and 1 is 6 
5 and 2 is 7 
5 and 3 is 8 



and 



6 from 6 is 1 
6 from 7 is 2 
5 from 8 is 3 



&c. 



* Cf . algebraic subtraction. 



32 CRAFTSMANSHIP IN MATHEMATICS 

"5 from 8?" is as effective a form of question as (and is 
much more elegant than) " 5 and what makes 8?". 

The Tables 

The child must learn the addition table to 9 + 9 perfectly. 
He must be able to say at once that, e.g., 9 and 8 is 17, He 
must also be able to say at once that 9 froni^rLiS-A and 
that 8jTom_jL7 is 9. In fact, the addition and subtraction 
tables should be learnt in close association. Very young 
children when learning to count, to add and subtract, will, 
of course, be shown how to find out that 8 + 311 and 
that 11 3 = 8, but the time must soon come when they 
can give those results pat, without calculation or thinking 
of any kind; and this means a great deal of sheer ding-dong 
work from which lower forms and classes can never escape. 
Never mind the charge of unintelligence; be assured that 
the people who make such a silly charge have never had to 
face the music themselves. Table accuracy is the one key 
tp_ accurate arithmetic. 

^EaclTsubtraction table is, of course, as already indicated, 
the mere complement of an addition table. For instance, 
the 4 times addition table begins 4 and 1 is 5 and ends 4 
and 9 is 13; the corresponding subtraction table begins 
4 from 5 is 1 and ends 4 from 13 is 9. Carry the addition 
tables to 9 + 9 and the subtraction tables to 18 9. 

How many repetitions are necessary to ensure permanent 
knowledge? All experienced teachers know that this varies 
enormously. It may be that only 10 repetitions are required, 
but it may be 500, according to the individual. Test, test, 
test, day by day. Do not waste the time of a whole class 
because further drill is necessary with a few. 

Helpful blackboard tests may be given in a variety of 
forms, e.g., 

9 + 6 = * 17 -4 = # 

6 + 8 = x 13 + 8 = x &c. 

In examples of this kind we have the germ of equations, 



ARITHMETIC: FIRST FOUR RULES 33 

as we had with the examples in brackets. Explain that x 
is a symbol for the number to be found. Call on a member 
of the class and point to the first x, then call on another member 
and point to the second x. But do not call on members in 
order. Keep every child in expectation. Call on Smith the 
shirker half a dozen times a minute. If the answers are not 
given at once, without any calculations, the tables are not 
known, and more drill is necessary. 

Draw a circle of small numbers on the board and have 
them added together, as they are pointed to. The answers 
must be instantaneous or the tables are not known. 

Make the children count forwards and backwards, by 
1's, then by 2's, then by 3's, &c. 

1, 4, 7, 10, 13, 16, &c. 
100, 96, 92, 88, 84, 80, &c. 

This sort of practice helps the tables greatly. 

But do not expect that, because a boy knows 7 + 6 is 13, 
he will therefore know that 27 + 6 = 33. Such extended 
examples require special practice, and the practice must 
be continued day by day until the boy knows at once that a 
7 added to a 6 always produces a 3. Similarly with sub- 
traction; a boy must be able to say at once that a 7 taken from 
a 6 always produces a 9. 

Write on the blackboard, say, a 7. " Let us add 6's." 
Smith? 13; Brown? 19; Jones? 25; dodging about the class. 
The response must be instant. Similarly with subtraction. 

Let your schemes for testing the tables be as varied as 
possible. Do not be satisfied as long as there is a single 
mistake. Do not forget that dull boys may require 10 times, 
perhaps 50 times, the practice that quick boys require. There 
must be no counting on fingers, no strokes, no calculations of 
ainy kindT 



So with the multiplication and division tables. Beginners 
are taught, of course, that multiplication is just a shortened 
form Q| a succession of additions, and dm 8 * 011 a _ shortened 
fofm^of a succession of subtractions. That fact grasped, 

fB~29n " " 4 



34 CRAFTSMANSHIP IN MATHEMATICS 

then come the tables, multiplication to 9 X 9 and division 
to 81 -f- 9. 

Do not be intelligently silly and teach a boy " to find 
out for himself " the value of 9 X 8 by making him set out 
9 rows of 8 sticks each and then count to discover 72. Make 
him learn that 9 X 8 ==72. _ When he begins multiplication 
and division, a few very easy concrete examples will be given 
him, to make the fundamental ideas clear. Then make him 
learn his tables, learn his tables. 

As with the addition and subtraction tables, write the 
multiplication and division tables side by side. The sign 
for " equals " may well be substituted for " is ". 

1X7=7 7's into 7 = 1 

2 X 7 = 14 7>s into 14 = 2 

3 X 7 = 21 7's into 21 = 3 



9 X 7 = 63 7's into 63 = 9 

Mental work: 



Seven threes? 
Three sevens? 
Sevens into twenty-one? 
Threes into twenty-one? 



Let the 3, the 7, and the 21 
hang together in all 4 
ways. 

And so on. 



Ask for the factors of such numbers as 42, 77, 28, &c. 

Blackboard Work: 

Write down a number consisting of 15 or 18 figures, 
and ask the class to give the products of successive pairs of 
figures, as rapidly as possible: e.g. 

371498652498. 
Answers: 

21, 7, 4, 36, 72, &c., 
Again: 

7 X 3 = x\ 3 X 7 = x\ 3 X x = 21; 7 X x = 21; x X 3 = 21; 

x X 7 = 21. 

21 21 

= x\ = x, 

37 



ARITHMETIC: FIRST FOUR RULES 35 

And so on. Point to an #, and call on a particular pupil for 
the answer. 

Mental work in preparation for multiplication and division 
sums: 

(3 x 7) + 1 = x. (3 x 7) + 2 = x (8 X 7) + 5 - x. 
3's into 22 = x. 3's into 23 = x. 8's into 61 = x. 

Ample practice in this type of example is necessary. The 
examples are of course one step beyond the simple tables; 
there are two operations, one in multiplication or division, 
one in addition or subtraction. Hence instantaneous response 
is hardly to be expected from slower children. But it is 
surprising how quickly the answers come from children 
who know their tables, who know that 8 X 7 = 56 and that 
56 + 5 = 61, though it is well to remember that a mental 
effort is required to keep in mind the first answer while it 
is being further increased or diminished. 

The 11 times table is hardly worth learning. The 12 times 
table may^ be ^ostponed^ until money sums are taken up. 
The 15 times is easy to learn and is useful for angle division. 
So is the 20 times table. Mental work on simple multiples 
is easy to provide, e.g. 18 X 9 = twice 9x9; (16 X 7) = 
twice (8x7). 

But when actually teaching the tables, it is a safe rule 
not to complicate matters by giving tips for exceptional cases. 
Do not, for example, tell a beginner that, when he is adding 
a column of figures, he should look ahead to see if two of 
them added together make 10. If he has to find the sum of 
4, 8, 3, 7, 5, teach him to say, 4, 12, 15, 22, Jj7^not to look 
ahead and to discover that 3 + 7 = 107~and then to say 
4, 12, 22, 27. Such a plan with beginners makes for in- 
accuracy. Good honest straightforward table work must 
come first. Short cuts may come later, when they may be 
more readily assimilated. 



36 CRAFTSMANSHIP IN MATHEMATICS 



Multiplication 

It is easy to make any average child who is well grounded 
in numeration and notation understand that 4 times 273 
means the sum of four 273's, i.e. 

273 
273 
273 
273 

and that therefore the answer is 

(4 times 200) + (4 times 70) + (4 times 3); 

and he sees readily enough that the teacher's shortened 
form 

273 200 70 3 

4 444 

1092 = 800+280+12 

But the slower child will not understand, though he will 
learn the ordinary rule of multiplication fairly readily. 

In teaching multiplication, the advisable succession of 
steps seems to be: 

(a) Easy numbers by 2, 3, and 4; no carrying; no zeros 
in multiplicand. 

(b) Easy numbers by 2, 3, and 4; no carrying; zeros in 
multiplicand. 

(c) Easy numbers by 2, 3 ... 9, with carrying; no zeros 
in multiplicand. 

(d) Easy numbers by 2, 3 ... 9, with carrying; zeros in 
multiplicand. 

(e) The same with larger multiplicands. 
(/) Multiplication by 10. 

(g) Multiplication by 2-figure numbers not ending in a 
zero. 



ARITHMETIC: FIRST FOUR RULES 37 

(h) Multiplication by 2-figure numbers ending in a zero, 
(i) Multiplication by 3-figure numbers, zeros varied. 

Be especially careful to show clearly the effect of multiplying 
by 10, viz. the shifting of every figure in the multiplicand one 
place to the left in the notational scheme, i.e. each figure is 
made to occupy the next-door position of greater importance. 
Then show the effect of multiplying by 100, by 1000, by 
20, 200, 6000, &c. Bear in mind that the work has particular 
value, inasmuch as ultimately it will lead on to decimals. 

From the outset, use the term multiplier and the term 
product, but let the difficult term multiplicand wait until the 
senior school stage. The term top line will do for juniors. 

When we come to ordinary 2-figure and 3-figure multi- 
pliers, which of the following processes is preferable, the 
first or the second? 

34261 34261 

43 43 



102783 1370440 

137044 102783 



1473223 1473223 

The first is the old-fashioned method; the second is newer and 
at present is popular. The second is often advocated because 
(1) it leads on more naturally to the rational multiplication 
of decimals, (2) it is preferable to multiply by the more 
important figure first, if only because the first partial product 
is a rough approximation to the whole product. 

The first reason does not appeal to me at all, for I am 
very doubtful about the allied method of multiplication of 
decimals. The second reason is undoubtedly a good one. 

Numerous tests of the comparative merits of the two 
methods have shown that the old method leads to a much 
greater accuracy than fE new, and to me that seems greatly 
to outweiglTthe advantage of the new method. Slower boys 
seem to have much more confidence in a method where they 
have to begin with both units figures, as they do in addition 



38 CRAFTSMANSHIP IN MATHEMATICS 

and subtraction. In any case I deny that the newer method 
is " more intelligent " than the old7 



Division 

Begin by instructing the children to write down in standard 
division form such little division sums as they know from 
their tables. Teach them the terms dividend, divisor, and 
quotient: we can hardly do without them. 

2(6 3\9 3|7 5|9 

3 3 2, and 1 over 1, and 4 over. 

Now teach them the use of the term remainder, and to write 
the letter R for it. 

5|9 4|6 

T, R 4 I, R 2 

Now 2-figure dividends, within the tables they know. 

4|36 8|47 9|79 

^9 , R 7 J, R 7 

Now 2-figure dividends beyond the tables they know. 



" 4's into 93? the tables do not tell us. Then let us take our 
4 times table further: 

10 x 4 = 40 

11 X 4 = 44 



23 x 4 = 92 

24 X 4 = 96 



" Evidently 4's into 93 are 23, and 1 R. Hence 

4|93 
23, R 1 



ARITHMETIC: FIRST FOUR RULES 39 

" But we need not have written out that long table; we may 
work in this way: 

" 4's into 9? 2 and 1 over; write down the 2. 

" By the side of the 1 over, write down the 3, to make 13. 

" 4's into 13? 3, and 1 over. Write down the 3. 

" The last 1 over is our Remainder. 

" But what does this mean? When we said 4's into 9 we 
really meant 4's into 90, and when we wrote down the 2, 
the 2 really meant 20. Here is a better way of showing it 
all, and we will write the figures of the answers above the 
dividend, instead of below it. 



4)93 

8_ = 80 = 20 times 4 
13 
12 = 3 times 4 



" First we took from the 93, 20 times 4, and had 13 left. 
" Then we took from the 13, 3 times 4, and had 1 left. 
" Altogether we took from the 93, 23 times 4, and had 
1 left." 

A little work of this kind will suffice to justify the process 
to the brighter children; a few will grasp it fully. The dullards 
will not understand it all; they want the clear-cut rule, and 
explanations merely worry them. 

Now consider an ordinary long division sum; say, divide 
45329 by 87. Let the children write out the 87 times table, 
to 9 X 87. 



1 X 87 = 87 

2 x 87 = 174 

3 x 87 = 261 



4 x 87 = 348 

5 X 87 = 435 

6 X 87 = 522 



7 X 87 = 609 

8 X 87 = 696 

9 X 87 = 783 



(In making a table like this note that 3 times = 2 times 



40 CRAFTSMANSHIP IN MATHEMATICS 

+ 1 time, 5 times = 3 times + 2 times, &c., and so save 
the labour of multiplying; only multiplication by 2 is necessary; 
all the rest is easy addition.) 



87)45329 

" 87's into 4? won't go: 4 is not big enough; put a dot 

over it. 
" 87's into 45? won't go: 45 is not big enough; put a 

dot over the 5. 
" 87's into 453? will go, because 453 is bigger than 87. 

How many times?" 

Look at the table, and take the biggest number (435) that 
can be subtracted from 453. The 435 is 5 times 87. Place 
the 5 over the 3 in the dividend, write the 435 under the 453, 
and subtract; the difference is 18. 

Bring down the 2 from the dividend, placing it to the right 
of the 18, making 182. Look at the table again, and take the 
biggest number (174) that can be taken from the 182, &c. 

"521 

87)45329 

435! : 

~-T82; 

174: 

"^9 

87 



" Thus we know that 87 is contained 521 times in 45329, 
and that there is 2 (the Remainder) to spare. 

" What is the biggest R we could have? Could it be 87? 
Why not?" 

Teach the children the usual verification check: multiply 
the divisor by the quotient, add R to the product, and so 
obtain the original dividend. 

(Do not forget, when introducing formulae later, to utilize 
the D, d, Q, and R. D = dQ or dQ + R.) 



ARITHMETIC: FIRST FOUR RULES 41 

Now pour a little gentle scorn upon making out a special 
multiplication table for every division sum: " We must 
give up such baby tricks ". But that leads us to what some 
beginners in division find very difficult how to tell the 
number of times a big divisor will go into one of the numbers 
derived from the dividend: 

69)342 

" Instead of saying * 69's into 342 ', we cut off the last figure 
of the 69 and of the 342 and say 6's into 34 instead. This 
seems to be 5, but the 5 may be too big, because of the carry 
figure; we find it is too big, so we try 4 instead. " 

Warn the children that if, after subtracting at any step, 
they have a difference greater than the divisor, the figure 
they have just put into the quotient is too small. Rub this 
well into the dullards. 

Warn them, too, that, above every figure in the dividend, 
they must place either a dot or a figure for the quotient. 

"341, R = 267. 
329)112466 

Similarly in short division except that the dots and figures 
go below: 

7)13259 

1894, R = 1. 



A dot is not a very suitable mark, owing to confusion with a 
decimal point; it is, however, in common use. / If no mark is 
used, figures get misplacedand errors made. The marks 
may be~ dropped later. ~~~ 

The advantage of the method of placing the quotient 
over jnstead of totheright of the dividend, jsjliat children 

write down noughts when these 



are reguired^. ' 

Eet division by factors stand over until the senior school. 
The calculation of the remainder is puzzling to beginners. 
Divide 34725 by 168. Suitable factors of 168 : 4 X 6 x 7. 



42 CRAFTSMANSHIP IN MATHEMATICS 

434725 units. 

8681 fours, R = 1 unit. 



u OUOA iuurs, r\ = JL unit. 

7 ' 1446 twenty-fours, R = 5 fours. 
' 206 one hundred and sixty-eights, R = 4 twenty-fours. 

Total Remainder = (24 x 4) + (4 X 5) + 1 

= 117. 
Quotient = 206, R =* 117. 

Avoid the Italian method, except perhaps with A Sets. 
With average children the method is productive of great 
inaccuracy. 

In fact, avoid all short cuts until main rules are thoroughly 
mastered. For instance if a boy has to multiply by 357, 
do not teach him to multiply by 7, and then multiply this 
first partial product by 50 to obtain his second partial product; 
it is simply asking for trouble. 

Of course, practised mathematicians do these things, but 
we have to think of beginners. Teach a straightforward 
method, and stick to it. Hints as to " neat dodges " and 
about " short cuts " are for the few, not for all. 



CHAPTER V 

Arithmetic : Money 

Money Tables 

No part of arithmetic is more important than the various 
manipulative processes of money. It is with us every day 
of our lives, and accuracy is indispensable. The ordinary 
money tables must be known, and thus more ding-dong work 
is necessary. This is mainly a question of a knowledge of 
the 12 times table. Five minutes' brisk mental work twice 



ARITHMETIC: MONEY 43 

a day will pay, sometimes with and sometimes without the 
blackboard, and sometimes on paper. 
Associate with the 12 times table: 



1 x 12 = 12 

2 X 12 = 24 

3 X 12 = 36 



I2d. = Is. 
24<*. = 2*. 
36 d. = 3s. , &c. &c. 



Day by day drill: 

SQd. = ? 83<f. = ? S4d. = ? 89<f. = ? &c. 
Pence in 7s. ? in 7s. 3d. ? in 9s. 9J. ? &c. 

and so every day until accurate answers up to 144*/. are 
instantaneous. If the boys are familiar with two definite land- 
marks in each " decade ", 20 and 24, 30 and 36, 40 and 48, 
&c., the" necessary additions for the other numbers of each 
decade are simple. 

Associate the farthings table with the 4 times table, and 
the shillings table with the 20 times table, which is easily 
learnt. 

Let every mental arithmetic lesson at this stage include 
simple addition and subtraction of money, especially the 
addition of short columns of pence. 

Elementary facts concerning the coinage should be 
associated with the money tables, and in this connexion do 
not forget guineas (which often figure in subscriptions and 
in professional fees) and Bank of England notes. 

At a later stage the boys should be taught such common- 
place facts about the coinage as every intelligent person 
ought to know, e.g. the nature of " standard " gold and 
silver, degrees of " fineness ", the nature of the present 
legalized alloy for " silver " coinage, the nqyket prices of 
pure gold and silver, the nature of bronze; alloys; tokens; 
the Mint. 

Reduction 

Reduction is not likely to give serious trouble, if the 
tables are known. The commonest mistake is to multiply 



44 CRAFTSMANSHIP IN MATHEMATICS 

instead of divide, or vice versa. Teach the boy to ask himself 
every time whether the answer is to be smaller or larger\ 
if smaller, to divide; if larger, to multiply. But " guineas to 
pounds ", and the like, is a type of sum that baffles the slow 
boy and requires special handling. 

Subtraction 

There is something to be said for avoiding, at first 
writing farthings in the usual fractional form and for giving 
them a separate column: 

* d. f. 

47 14 6 1 

21 19 4 3 

25 15 1 ~2 



let the children use the fractional forms a little later, when 
they may be made a useful introduction to fractions. The 
alternative is to omit farthings altogether in the early stages. 

Multiplication 

How is this to be done? Whatever method is adopted, 
a percentage of inaccurate answers seems almost to be in- 
evitable. We set out a sum by each of the 4 methods commonly 
used. Multiply 7, 15$. 10W. by 562. 

I. s. d. 

7 15 10J 

10 

77 18 9 = 7 15 10J X 10 

10 

779 7 6 = 7 15 10J X 100 
5 



3896 17 6 = 7 15 10J x 500 

15 11 9 = 7 15 10J x 2 

467 12 6 = 7 15 10J X 60 

4380 1 9 = 7 16 10J X 562 



ARITHMETIC: MONEY 



II. 



662 
16 



20| 84305. 

421,105. Od. 



12[6620</. 

20 468^. 



2562/z. 



23, 85. 4d. 



1228U. 



235. 5d. 



45 



7 








X 


662 = 


3934 











16 





X 


662 = 


421 


10 











10 


X 


562 = 


23 


8 


4 








t 


X 


562 = 


i 


3 


5 


7 


16 


10* 


X 


562 = 


4380 


1 


9 





III. 



562 at 7, 155. 

7 


lojd. 






3934 





= 


7 








X 


562 


* 


ofl 


281 





= 





10 





X 


562 


t 


of 10/- 


140 


10 


= 





5 





X 


562 





of 5/- 


17 


11 


3 =r 








7J 


X 


562 


i 

tv 


of 5/- 


7 





6 = 








3 


X 


562 




4380 


1 


9 = 


7 


15 


iot 


X 


562 



IV. 



7, 165. 10H x 562 
= 7-79375 x 562 
= 4380-0875 
= 4380, 15. 9d. 



My own experience, and this corresponds to the results of 
many inquiries, is that the second method produces the best 
results; then the third (" practice ") method, provided that 
pupils have been well drilled in aliquot parts (though some 
always seem to find division more difficult than multiplication); 
then the first method. The last is a good method for older 
pupils who have learnt to decimalise money readily, but 
not for younger pupils or for slower older pupils. 

The ordinary method (the first method) is curiously 
productive of errors; in the course of a long experience I 
have never known a whole class, without exception, get a 
sum right by this method, even after they had had several 
months' practice. The second method generally leads to 
untidy and unsystematic marginal work. This marginal 



46 CRAFTSMANSHIP IN MATHEMATICS 

work should be made an integral part of the working of the 
sum, and should not be looked upon as scrap work. 



Division 

Dr. Nunn's suggestion that the process of working may 
be set out in the following way may well be followed. All 
pounds are kept in one vertical column, shillings in another, 
and so on. It is very neat and compact. Allow plenty of 
space across the paper. Example: Divide 3541, 14s. 
by 47. 

s. d. f. 

75 7 1 1 



47)3541 
329 


14 9 2 
->320 ( ->60 ,-88 


251 


334 


69 


90 


235 


329 


47 


47 


16 X 20- 


5 X 12- 


22 X 4- 


43 R 



Answer: 75, 7s. \^d. and 43 farthings over. 

Other methods have been devised, but this old method is 
probably best and most readily learnt. 

To ensure a full understanding of the nature of the 
" remainder " a sum like the above should be followed up 
by two others: 

1. Take 43 farthings from the dividend; then divide 

again. 

2. Add 4 farthings to the dividend; then divide again. 

Even slower boys can usually explain the (to them) rather 
surprising new answers. 



WEIGHTS AND MEASURES 47 

CHAPTER VI 

Weights and Measures 

Units and Standards 

Consider what weights and measures are used, to what 
extent, and how, in practical life. Teach these, and teach 
them well, and let all the rest go. A coal merchant concerns 
himself with tons, cwt., qr., never with Ib. and oz.; a grocer 
with cwt., qr., Ib., oz., never with drams and rarely with tons; 
a farmer with acres, quarter-acres, and perches; a builder 
with yards, feet, and inches; a surveyor with chains and links; 
and so generally. The completer tables of weights and 
measures are generally given in the textbooks as a matter 
of convenience, but, in practical life, the whole table is 
seldom wanted by any one person. A teacher who gives boys 
reduction, multiplication, or division sums, say, from tons 
to drams (or even to ounces), or from square miles to square 
inches, is simply proclaiming aloud his incompetence: perhaps 
he is the slave of some stupidly written textbook; certainly 
he is lacking in judgment. The main thing is to make the 
boys thoroughly familiar with the few weights and measures 
that are commonly used, and to give them a fair amount of 
practice in the simpler transformations of comparatively small 
quantities; and to let all the rest go. 

Teach clear notions of units and standards. Show how 
unintelligent we British people have always been in our 
choice of units. We have, for instance, determined our inch 
by placing three grains of barley in line; we have selected 
our foot, because 12 of the inches roughly represent the 
length of a man's foot; we have determined our smallest 
weight (the grain), by adopting the weight of a dried grain 
of wheat. That such things vary enormously did not trouble 
our forefathers at all. Tell the boys that at one time the 
French people had similar unsatisfactory weights and measures 



48 CRAFTSMANSHIP IN MATHEMATICS 

but that now they have changed to a system much more 
rational. 

Let the various tables be learnt and learnt perfectly. 



Weight 

1. Avoirdupois (not used for the precious metals). Let 
the table to be learnt include the oz., Ib., qr., cwt., ton. 
Note that the standard weight is the pound, which consists 
of 7000 grains. (A dried grain of wheat, though roughly a 
grain in weight, is not, of course, a standard. A grain is 
1/7000 part of a pound.) 

Teach the stone as a separate item: normally 14 lb., but for 
dead meat, 8 Ib. 

Give easy sums for practice in: 

(1) tons, cwt. y qr. (coal and heavy goods). 

(2) cwt., qr., Ib. (wholesale grocery). 

(3) cwt., st., Ib. (wholesale meat purchases). 

(4) Ib., oz. (retail grocery and meat). 

Note that an oz. of water or any other fluid is an avoirdupois 
ounce, like the ounce of any common solid, and contains 
7000/16 or 437 grains. 

In making up arithmetical examples, utilize as far as 
possible the quantities (sacks, bags, chests, &c.) representing 
the unit purchases of tradesmen and others, though the 
problem given will often depend on the locality. For instance, 
problems on crans and lasts of herrings, or on trusses of hay 
or straw, would be quite inappropriate in big inland towns. 
The teacher should, for problem purposes, make a note of 
points like the following: weight of a chest of tea, f cwt.; 
sack of coal or of potatoes, 1 cwt.; bag of flour, 1J cwt.; 
bag of rice, 1 cwt.; truss of straw, 36 Ib.; truss of new hay, 
60 Ib., of old hay 56 Ib.; a brick, 7 Ib.; 1000 bricks, 3 tons; 
100 Ib. of wheat produces 70 Ib. of flour which produces 
91 Ib. of bread; and so on. Everyday quantities of this kind 



WEIGHTS AND MEASURES 49 

give a reality to problems in arithmetic that even the non- 
mathematical boy appreciates. 

2. Troy, used by jewellers. Let the table to be learnt 
include the grain, dwt., oz., Ib. It is important to remember 
that the Troy ounce is heavier than the common (avoirdupois) 
ounce, since it contains 480 grains, as against 437^. But 
the Troy pound is lighter than the common pound, since it 
contains only 12 ounces and therefore 5760 grains, as against 
7000 in the avoirdupois Ib. 

3. Apothecaries'. The old weights have gone out of use. 
Drugs are generally used in very small quantities, and the 
basic weight is the grain (the grain is a constant weight for 
all purposes). A quantity of drugs weighing more than a 
few grains is expressed as a fraction of an ounce avoirdupois. 

N.B. Ignore the avoirdupois dram (^ oz.) and the 
druggists' old scruple and drachm weights. The dram and 
drachm were not the same. 



Length 

Let the main table to be learnt include the in., ft., yd., 
pole, fur ., mile, and let the link, chain, and fur. be included in 
a separate table. Remind the boys that the chain is the 
length of a cricket pitch. 

Give easy sums for practice in: 

(1) yd., ft., in. (builders, &c.). 

(2) poles, yd., ft. (farmers, &c.). 

(3) miles, chains, links (surveyors). 

(4) miles, yd. (road distances, &c.). 

Measures that may be drawn from practical life for 
problem use are almost innumerable. The sizes of battens, 
deals, and planks will be learnt in the manual room; notes 
of the sizes of other materials used by builders slates, 
glass, door-frames, &c., &c. may be made from time to time; 
size of an ordinary brick, 8f " X 4" X 2" (note the \" all 

(E291) 5 



60 CRAFTSMANSHIP IN MATHEMATICS 

round for jointing), square tile, 9f" X 9f " X 1" or 6" X 6" 
X 1"; machine-printed wall-paper, 11 \ yd. X 21"; hand- 
printed, 12 yd. X 21"; French, 9 yd. X 18"; sheets of 
paper, foolscap, 17" X 13|" (see Whilaker for other sizes); 
bound books, foolscap 8vo, 6f" X 4|" (see Whitaker)\ 
skein of yarn = 120 yd., hank = 7 skeins; railway gauge 
4' 8|" (12' of roadway for single track, 23' for double); equator, 
24,902 miles; polar diameter, 7926 miles; fathom, 6'; knot, 
6080' (40 knots == 46 miles). These are only a tithe of the 
everyday measurements that may be used for making up 
problems. Such problems are far more valuable than the 
hackneyed reduction sums given in the older textbooks. 

N.B. The ell, league, and such foreign lengths as the 
verst, may be ignored. The cubit is worth mentioning. 

Area 

Let the table to be learnt include the sq. in., ft.,, yd. pole, 
the rood, the acre, sq. mile. It is useful to remember that an 
acre = 10 sq. chains, or a piece of ground 220 yd. X 22 yd., 
or a piece just about 70 yd. square. 

Give easy sums for practice in: 

(1) sq. miles, acres (areas of counties, &c.). 

(2) ac., ro., sq. poles (farmers, &c.). 

(3) sq. yd., sq.ft., sq. in. (builders, &c.). 

Familiar areas for problem purposes: Lawn tennis 
court, 78' X 36' or 78' X 27'; Association football ground, 
120 yd. X 80 yd.; Rugby, 110 yd. X 75 yd.; croquet lawn, 
105' X 84'; Badminton court, 44' X 20'; &c. 



Volume 

Table: c. in., ft., yd. 

Let sums for practice be of the simplest, e.g. the number 
of cubic yards of earth excavated from a trench; the number of 



WEIGHTS AND MEASURES 51 

cubic feet of brickwork in a wall; the cubic capacity of a 
room or of a building; the volume of the Earth in. cubic 
miles. 

Capacity 

1. Liquids. Table: gill, pt., qt., gall. Casks have a 
variety of names: barrel of ale 36 gall.; hogshead of ale 
54 gall., of wine = 63 gall., &c. A wine bottle = gall.; 
Winchester quart = \ gall. 

2. Dry Goods (corn, &c.). Table: peck, bushel, quarter. 
The quarter-peck is called a " quartern "; the half-peck is 
the equivalent of a gallon. The gallon is a kind of link between 
the liquid and dry measures. 

There is now a tendency to substitute weight for measure. 
Problems on capacity reduction are hardly worth doing, 
except small problems that may be done mentally. But 
problems involving transformations between capacity and 
weight are common, and ample practice is necessary. N.B. 
1 gall, of water weighs 10 Ib. " A pint of pure water weighs 
a pound and a quarter." 

Liquid medicine measure (mainly solutions in water). 

Table: 60 minims = 1 fluid drachm. 

8 fluid drachms = 1 fluid ounce. 
20 fluid ounces = 1 pint. 

The fluid ounce is the same as the common (avoirdupois) 
ounce, and therefore weighs 437^ grains. But it contains 
480 minims, and therefore a minim weighs rather less than 
a grain. The minim may be thought of as a " drop ", though 
of course drops vary greatly in size. 

Doctors' prescriptions may be discussed, rather than 
sums worked. If a solid drug is prescribed, the amount is 
expressed in grains or in fractions of an ounce; if liquid, then 
minims, drachms, or ounces. 



52 CRAFTSMANSHIP IN MATHEMATICS 

Time 

The second, minute, hour, day, week, give little trouble. 
The variable month requires careful explanation. Teach 
the doggerel " Thirty days hath September ", &c., or furnish 
some alternative mnemonic. Explain " leap " year and its 
determination. 

Few problems of reduction are necessary, and these should 
be easy. A few on the calendar are advisable, and a few 
dealing with speeds. 

Useful Memoranda 

Other useful memoranda for problem-making. (The quan- 
tities are approximations only and should be memorized. 
They are useful when closer approximations have to be 
evaluated): 

1 cubic foot of water 6J gall. 62 J Ib. 
1 cubic inch of water = 252 grains. 

A common cistern 4' x 3' X 2* ' = 30 c. ft. 187 gall. = ton. 
1 ton of water = 36 c. ft. = 224 gall. 
1 gallon of water = 277| c. in. = 10 Ib. 

1 ton of coal occupies about 40 c. ft. (hence 25 tons need a 
space 10' X 10' X 10'). 

Wall Charts 

A few permanent charts are useful on a wall of the class- 
room where weights and measures are taught: an outline 
plan of (1) the town or village showing the over-all dimen- 
sions, length, breadth, and area; (2) the school-site and 
buildings; (3) the school itself; (4) the actual classroom. 

(5) Diagram to scale to show that 5-| yd. X 5|- yd. = 
30 J sq. yd. (often used for a first lesson in fractions). 

(6) A chart giving the weights of a few familiar objects in 
and about the school, and the capacities of a few others. See 
that these charts are used and known. 



WEIGHTS AND MEASURES 53 



The Metric System 

Some knowledge of this system is necessary, if only 
because of the work in the physical laboratory. The be- 
ginner may be shown a metre measure side by side with a 
yard measure, and simply be told that it is rather longer, 
and had its origin in France. As the boy goes up the school 
he will learn that its length is about 39-37 in., and is the 
measured fraction of a quadrant of the earth's surface. A 
little later still, he will be told how the French measured 
the actual length of an arc of one of their meridians, and 
how they determined the latitude of each place at the end of 
the arc. This easy astronomical problem is usually worked 
out in a Fifth Form geography lesson. 

The cubic decimetre and the litre, the cubic centimetre 
and the gram, are, as derivations of the initial metre, always 
a source of interest to boys. 

The boys should memorize the few usual approximate 
equivalents between the British and metric systems, e.g. 
1 metre = 39-37 in.; 1 kilogram = 2-2 lb.; 1 litre = 1-76 
pints; 1 gram = 15-43 grains; 1 are = -$ acre. With these 
they can quickly estimate quantities in terms of metric units. 
For instance, a Winchester quart will hold 4/1-76 litres 
= 2-27 litres = 2270 c. c.; 1 hectare = 2 acres; and so on. 

But do not forget to enter a defence in favour of our own 
system of weights and measures, if the metric system is 
advocated on purely scientific grounds. Sixth Form boys 
are always interested in this. In the first place, the metre 
was not measured accurately; in the second place, it is a 
local and not a universal unit; it depends upon the length 
of a particular meridian in a particular country. The meridian 
the French measured was an ellipse, not a circle, and not 
a true ellipse at that. Had they utilized the polar axis (a 
fixed length) instead of a meridian (a variable length), their 
unit would have been more scientific, for it would have been 
universal, and it could have been measured more accurately. 



54 CRAFTSMANSHIP IN MATHEMATICS 

The length of the polar axis is very nearly 500,500,000 in., 
so that the inch already bears a simpler relation to the polar 
axis than the metre does to its own meridian quadrant. If we 
adopted a new inch, viz. 1/500,000,000 of the polar axis, 
it would make but a very slight change in our linear measure- 
ments, and then, curiously enough, a cubic foot of water 
would weigh almost exactly 1000 oz. (instead of 997). Our 
present ounce weight would have to be increased by only 
Y- part of a grain! Moreover, the new cubic foot would 
contain exactly 100 half-pints. Such a new system would 
be incomparably more scientific than the metric system. 

Thus the opponents to the adoption of the metric system 
have sound arguments to support their views. The metre 
has on its side the virtue of being the basic unit of a con- 
venient and simple system; but scientifically it is a poor 
thing. 

There is no need for the boys to learn the metric tables. 
But they should learn the three Latin prefixes deci, centi, milli, 
and know that these represent fractions; and the three Greek 
prefixes deca, hecto, kilo representing multiples. These learnt 
thoroughly, the tables as such are unnecessary. But with 
three or four exceptions the multiples and sub-multiples are 
hardly ever wanted. 

The best exercises on the metric system are those based 
on laboratory operations. 



CHAPTER VII 

Factors and Multiples 

The term " factor " and " multiple " should be used 
when the tables are being taught, though without formal 
definition. " 3 X 7 = 21; we call 3 and 7 factors of 21." 

Give me a factor of 6? 3; another? 2; a factor of 30? 2; 



FACTORS AND MULTIPLES 55 

another? 3; another? 5, A multitude of people means 
many people, and a multiple of a number means a bigger 
number containing it many times, though " many " may not 
be greater than 2. Now think of your 5 times table. Give 
me a multiple of 5? 15; another? 30; another? 35. After a 
little of this work, the terms factor and multiple will become 
part of the boys' familiar vocabulary. " Common " factor 
and multiple will come later. One idea at a time. 



Tests of Divisibility 

Prime Factors. Tests of divisibility for 2, 3, 5, and 10^ 
may readily be given in the Junior School or Department; 
those for 4, 8, 9, 11, 12, 25, 125, a year or two later. At first, 
give the rules dogmatically. 

" A number is divisible by 2 if it is an even number. 

3 if the sum of its digits is 

divisible by 3. 

5 if it ends in a 5 

10 if it ends in a 0." 

Justification, not " proofs ", of such rules is commonly 
given in Form IV. The reasoning, which is quite simple, 
depends on the principle that a common factor of two 
numbers is a factor of their sum or their difference. Never 
mind the general proof; at this stage merely justify the 
principle by considering a few particular instances, and 
these readily emerge from the multiplication table; for 
instance: 



5 fours = 20 
7 fours = 28 



12 fours = 48 



9 fives = 45 
7 fives = 35 
2 fives = K) 



We know that 5 fours added to 7 fours make 12 fours, i.e. 

4 is a factor of 20 and of 28, and is also a factor of 20 + 28. 

Again, we know that 7 fives from 9 fives is 2 fives, i.e. 



66 CRAFTSMANSHIP IN MATHEMATICS 

5 is a common factor of 35 and 45, and is also a factor of 
45 - 35. 

Divisibility by 2. Consider any even number, say 754; 
754 = 750 + 4. Since 2 is a factor of 10 and therefore of the 
multiple 750, and is also a factor of 4, it is, by our rule, a 
factor of 750 + 4 or 754. 

Divisibility by 5. Consider any number ending in 5, 
say 295; 295 = 290 + 5. Since 5 is a factor of 10 and there- 
fore of the multiple 290, and is also a factor of 5, it is, by 
our rule, a factor of 290 + 5 or 295. 

Divisibility by 3. Consider any number, say 741. 

741 = 700 -}- 40+1 

= (100 x 7) + (10 X 4) + 1 
= (99 X 7) + 7 + (9 X 4) + 4 + 1 
= (99 X 7) + (9 X 4) -{- 7 -f 4 + 1 
= (99 X 7) + (9 X 4) + 12. 

Now 3 is a factor of 9 and therefore of all multiples of 9; 
it is also a factor of 12. Since 3 is a factor of 99 X 7 and of 
9x4 and of 12, it is a factor of their sum, i.e. of 741. 
Hence, &c. 

A formal proof of the principle used should be associated 
with the algebra later. 

The justification of the rule for 4 and 25, 8 and 125, 
and 9 is equally readily understood, but that for 11 is a 
little more difficult. 

Primes and Composite Numbers 

Quite young boys quickly see the distinction between a 
prime and a composite number and are always interested 
in the sieve of Eratosthenes. 

Third Form boys should be made to memorize the squares 
of all numbers up to 20; 13 2 = 169; 17 2 = 289; &c. (The 
squares of 13, 17, and 19 must really be learnt; 14 2 , 16 2 , 
and 18 2 can be mentally calculated in a second or two, if 
forgotten.) Then give a little mental practice in extracting 



FACTORS AND MULTIPLES 57 

square roots: of 81? of 256? of 361? (Mention that the root 
sign (vO we use is merely a badly written form of the initial 
letter R.) 

Make the class write down, in order, the successive pairs 
of factors of, say, 36: 

2 X 18 

3 x 12 
4x9 
6X6 

I 9 X 4 
112 x 3 
\18 X 2 

Then point out that the second column is the first column 
reversed, and that the 3 lower horizontal lines are the 3 
upper horizontal lines reversed. Hence when we have to 
write dowu the factors of 36, we need not proceed beyond 
the fourth line, viz. 6x6, for then we already have all the 
factors; and the 6, the last trial number, is \/36. The boys 
can now appreciate the common rule: When resolving a 
number into factors, it is unnecessary to carry our trials beyond 
its square root, unless the number is not a perfect square, 
and then it is advisable to consider the next square number 
beyond it. 

For instance, write down all the factors of 120; 120 is 
not a square number, but the next square number is 121, 
the square root of which is 11. Hence we need not proceed 
with our trial numbers beyond 11, but as 11 does not happen 
to be a factor, we do not proceed beyond 10. Thus by trial 
we find that 2, 3, 4, 5, 6, 8, and 10 are factors; and, dividing 
120 by each of these, we obtain other factors which pair off 
with them. The 14 factors of 120 are 

2 3 4 5 6 8 10 
60 40 30 24 20 15 12 

The next step is to teach factor resolution by trials of 
prime numbers only. This causes no additional difficulty, 
but the boys should recognize at once all the prime numbers 



68 CRAFTSMANSHIP IN MATHEMATICS 

up to, say, 41 (1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 
37, 41). 

" Write down the factors of 391. The next square number 
beyond is 400. Hence we need not carry our trials beyond 
-V/400 or 20. By trial we find that the lowest factor is 17. 
By division we obtain 23, which we recognize as another 
prime number. Thus 17 and 23 are the only factors. " 

I have known a C Set of a Fourth Form become expert in 
factorizing 3-figure numbers, after one lesson. It is work 
that most boys like. 

" Express 360 as a product of factors which are all prime. 
Divide by the lowest prime number, 2 if possible; and 
again and again if necessary; then by the next prime, 3, 
if possible; and again and again, if necessary. Then by 5, 
if possible; then 7; then 11; and so on. 

360 = 2 x 180 

= 2 X 2 X 90 

= 2x2x2x45 

= 2x2x2x3x15 

= 2x2x2x3x3x5. 

Obviously we now have all the factors, though our trial 
division did not proceed beyond 3. 

" A neater way of writing down the prime factors of 350 is 

2 8 x 3 a x 5. 

The little 3 at the top right-hand corner of the 2 shows 
the number of twos and is called an index. 

" Express 18900 as the product of factors which are all 
prime. 

18900 = 2x2x3x3x3x5x5x7 
= 2 2 x 3 3 X 5 2 x 7. 

We read, 2 squared into 3 cubed into 5 squared into 7. 
An index serves as a useful means of shortening our written 
work." 

At this stage two or three minutes' brisk mental work 
occasionally will help to impress upon the pupils' minds the 



FACTORS AND MULTIPLES 59 

values of the lower powers of the smaller numbers: 2 3 , 2 4 , 
2 5 , 3 2 , 3 3 , 3 4 , 4 2 , 4 3 , 4 4 , 5 2 , 5 3 , 5 4 , &c. 



Common Factors 

" Give me a common factor of 36 and 48: 2; another? 
3; another? 4; another? 6; another? 12. Which is the 
greatest of these common factors? 12. We call 12 the Greatest 
Common Factor of 36 and 48. If we write down the prime 
factors of the different numbers, we can almost see the G.C.F. 
at once. 

36 = 2x2x3x3. 

48 = 2x2x2x2x3. 

Evidently 2 is a common factor of both numbers, and 
another 2, and a 3. Hence, the G.C.F. == 2 x 2 x 3 == 12, 
i.e. 12 is the greatest number that will divide exactly into 36 
and 48." 

" It is neater to write down the factors in the index form. 
What is the G.C.F. of 540, 1350, 2520? 

540 = 2 2 x 3 3 x 5 1 . 
1350 = 2 1 X 3 3 X 5 2 . 
2520 = 2 3 X 3 a X 5 1 X 7 

We see that 2, 3, and 5 are factors common to all three 
numbers; from the indices we see that one 2, two 3's, and 
one 5 are common. Hence the G.C.F. is 2 1 X 3 2 X 5 1 = 90. 
Note that we write down each prime factor that is common 
and attach to it the smallest index from its own group. " 

However clear the teaching, I find that there is usually 
a small number of slow boys who are puzzled by the index 
grouping. Hence in lower Sets the extended non-indexed 
groups of factors are preferable. Always sacrifice a neat 
method if it leads to puzzlement and inaccuracy. 



60 CRAFTSMANSHIP IN MATHEMATICS 



Common Multiples 

' Give me a multiple of 5: 25; another? 35; another? 
55. Give me a multiple of 3: 21; another? 15; another? 
60. Give me a common multiple of 3 and 5: 60; another? 
15; another? 30. Which is the least of all the common 
multiples of 3 and 5? 15; i.e. 15 is the smallest number into 
which 3 and 5 will divide exactly. We call it the Least 
Common Multiple^ 

" Find the L.C.M. of 18, 48, and 60." 

Write down the numbers as products of their factors, 
expressed in primes. 

18 = 2 X 3 X 3. 

48 2x2x2x2x3. 
60 =2x2x3x5. 

The L.C.M. , being a multiple of the three given numbers, 
must contain all the factors of the numbers, but it must 
not contain more, or it will not be the least common multiple. 
" 1. The L.C.M. has to contain all the factors of 18; 
write them down as part of the answer: 

L.C.M. = 2 x 3 x 3 x 

" 2. In order that the L.C.M. may contain all the factors 
of 48, it must include four 2's and one 3. We have already 
written down one 2; hence we must write down three more. 
As we have already written down two 3's, another is not 
necessary. Hence, 

L.C.M. =2x2x2x2x3x3x 

" 3. In order that the L.C.M. may contain all the factors 
of 60, it must include two 2's, one 3, and one 5. We already 
have two 2's, and a 3, but no 5. Hence we must include 
a 5. ^ < ^ ^ ^ ^ ^/ 

L.C.M. = 2x2x2x2x3x3x5 
= 720. 



FACTORS AND MULTIPLES 61 

" It is neater to write down the factors in the index form. 
What is the L.C.M. of 54, 72, 240?" 

64 = 2 X 3 3 . 
72 = 2 3 X 3 2 . 
240 = 2 4 X 3 1 X 6 l . 

We mayjvritc down the L.C.M. at once^, by writing downjwerx 
one of the prime^ factors and attacEmg to_gach the greatest 
index ot its group: 

L.C.M. = 2 4 x 3 s x 5 
= 2160 

It is well to provide pupils with some little mnemonic, 
to enable them to keep in mind that: 

the smallest index concerns th^ greatesJ^C^. 
and the greatest index concerns the fegg^CJVI^ 

There is much to be said for using the terms greatest 
and smallest (or least) in arithmetic, and the terms highest 
and lowest in algebra. The former terms are obviously 
correctly applicable to magnitudes. Beginners do not 
find it easy to appreciate the exact significance of highest 
and lowest. The term measure is best avoided. The dis- 
tinction between it and factor is a little subtle for boys. 

The old-fashioned division methods of G.C.F. and L.C.M. 
are cumbrous and unnecessary, and slower boys never un- 
derstand the processes, the formal " proofs " of which are 
quite difficult enough for Fifth Forms. Numerical illustrations 
of the principle that a common factor of two numbers is a 
factor of their difference may be utilized to justify the ordinary 
G.C.F. procedure, if the procedure itself is considered 
necessary. 



62 CRAFTSMANSHIP . IN , MATHEMATICS 

CHAPTER VIII 

Signs, Symbols, Brackets. First 
Notions of Equations 

Terminology and Symbolism 

Mathematical terms should always be used with precision; 
then formal definitions in all early work will be unnecessary. 
Sum, difference, product, quotient are terms which should be 
quite familiar even to Juniors; they are the A, B, C of the 
whole subject. So should the signs + , , X, and -. 
Multiplicand and multiplier, dividend and divisor should also 
become current coin at an early stage, though there is difference 
of opinion about the first term in this group. I am not quite 
sure about subtrahend and minuend, even in the senior school; 
they are commonly confused. If we bear in mind the English 
significance of the Latin -nd- (gerundive), the -nd terms 
can be explained in a group. 

Multiplicand, is a number that has to be multiplied. 
Dividend is a number that has to be divided. 
Subtrahend is a number that has to be subtracted. 

And of course minuend is a number that has to be minus-ed 
or reduced, but boys will confuse minuend and subtrahend. 
If the terms are used let subtrahend come first, and minuend 
a good deal later. 

As for the division sign -f-, hammer in the fact that the 
dots stand for numbers, that when we write, e.g., |- we mean 
4 divided by 5, and that we might write out our division table, 



either 



5-1 = 5 
10 -f- 2 = 5 
15 ~ 3 = 5 



Y = 



since both mean exactly the same thing. 

Algebraic letter symbols may be introduced at a very 



SYMBOLS, BRACKETS, EQUATIONS 63 

early stage. (Do not look upon arithmgtjc^giijd algebra^as 
cUstantjcouin&, but as twin brothers, children to be brought 
up together.} Begin with the simple consideration of lengths 
and areas. Establish by a few numerical examples that the 
area of a rectangle may be determined by multiplying length 
by breadth. Select rectangles whose sides are exact inch- 
multiples, ignoring all fractions until later. Then introduce 
the notion of a " formula " a convenient shorthand means 
of keeping an important general arithmetical result in our 
mind. " We have found that however many inches long, 
and however many inches broad, a rectangle is, the area in 
square inches is equal to the product of the inches length 
and the inches breadth. It is easy to remember this by taking 
the first letter of the word length (/), of the word breadth 
(6), and of area (A), and writing the result so: 

/ x b = A. 

But we generally save time by writing Ib = A, omitting the 
multiplication sign. Always remember that when in algebra 
two letters are written side by side, a multiplication sign 
is supposed to be between them. Instead of the letters /, 
b y and A, any other letters might be used." 

Rub in well the principle taught, giving a few simple 
evaluations. 

Now consider a square area, / X /, or m X m\ II or mm. 
(Distinguish between inches square and square inches.) 

Follow this up with cases of rectangular solids, and 
establish such formulae as V = I X b X h = lbh\ then the 
cube, V = aaa. 

" When we were working factors, we adopted a plan for 
shortening our work. Instead of writing 4 X 4 X 4, we wrote 
4 3 , the little 3 at the top right-hand corner (which we called 
an index) showing the number of 4's to be multiplied together. 
So in algebra. 

aa may be written a 2 , 
aaa may be written 3 . 

Then what does a 5 mean? a 3 i 2 ?" And so on. 



64 CRAFTSMANSHIP IN MATHEMATICS 

Avoid all difficult examples at this stage. The main thing 
is to teach the new principle. Keep the main issue clear. 
Let hard examples wait. 

Brackets 

" A pair of brackets is a sort of little box containing 
something so important that it has to receive special attention. 
The brackets generally contain a little sum all by itself. If 
I write 

9 + (7 + 3) 
or 9 -f (7 - 3), 

I mean that the answer to the little sum inside the brackets 
has to be added to the 9. If I write 

- (6 + 2) 

or 9 - (5 - 2), 

I mean that the answer to the little sum inside the brackets 
has to be subtracted from the 9. 

" Now I will work out the four sums: 

9 + (7 + 3) = 9 + 10 = 19. 

9 + (7 - 3) = 9 -f 4 = 13. 

9 - (5 + 2) = 9 - 7 = 2. 

9 _ (5 _ 2) = 9 - 3 - 6. 

Are the brackets really of any use? Let us write the same 
sums down again, leaving the brackets out, and see if we 
get the same answer: 

9 4- 7 + 3 = 19 
9 + 7 - 3 = 13 
9 - 5 -f 2 = 6 
9 - 5 - 2 = 2. 

The first two answers are the same, the last two are not. But 
look at the last two again. It looks as if they had been changed 
over. Thus 

9 (5 + 2) is the same as 9 5 2, 
and 9 (5 - 2) is the same as 9 - 5 + 2." 



SYMBOLS, BRACKETS, EQUATIONS 65 

With a few easy examples like this, we are in a position to 
justify the rule that a -f sign before a bracket does not affect 
the + and signs within, but that a sign before a bracket 
has the effect or converting + and signs within to and 
f~~ respectively^ Thus you are able to give the rule and to 
justifyit! That is enough at present. Give enough easy 
examples to ensure that the rule is known and can be applied 
with certainty. " Proof " should play no part at this early 
stage. The algebraic minus sign comes later. 

Now show the effect of a multiplier. 

4(6 -f- 3) = 4 X 9 == 36. 

" We might have multiplied the two numbers separately 
in this way, 

4(6 + 3) - 24 + 12 = 36,* 

and when the brackets contain both letters and numbers we 
must do it in that way: 

5(N -f 3) = 5N + 16 
for we cannot add N to 3." 



First Notions of Equations 

Simple equated quantities. For convenience at this early 
stage we may call the following an equation: 

7 + 5 = 21 - 9. 

Establish the fundamental fact about an equation that we 
may add to, subtract from, multiply, or divide each side of 
an equation by any number we like, provided that we use 
the same number for both sides. Give several examples, 
to illustrate each of the four operations. To enable the class to 



* A repetition of the sign of equality in the same line should never be allowed 
in school practice; it is almost always ambiguous. We do it sometimes in this book 
merely to save space. 

( E 291 ) 6 



66 CRAFTSMANSHIP IN MATHEMATICS 

see the operations more clearly, put the original quantities 
in brackets. Thus: 

(7 + 5) + 4 = (21 - 9) + 4 
(7 + 5) - 4 = (21 - 9) - 4 
4(7 + 5) = 4(21 - 9) 
7 + 5 21-9 



Do not talk about " proofs "; you are merely verifying 
particular instances, to enable the boys to see that your rules 
are not arbitrary but are based on reason. A little practice 
in such easy examples as the following may usefully follow. 

5 times a certain number is 65. What is the number? 

" We have to find a certain unknown number. Let us call 
it N. The sum tells us that 

5N == 65. 

Divide each side of the equation by 5; then, N = 13, the 
number we require." 

The class is not quite ready for such an example as the 
following, but they can follow out their teacher's reasoning, 
and their appetite is whetted. 

Divide 32 into two parts, so that 5 times the smaller is 
3 times the greater. 

" The two parts added together must make 32, so that 
one part taken from the 32 must give the other. 

Let S stand for the smaller number. 
Then 32 S must represent the greater. 
The sum tells us that 

6 times the smaller == 3 times the greater. 

So we may write 

6S = 3(32 - S). 



SYMBOLS, BRACKETS, EQUATIONS 67 

Removing the brackets, by multiplying by 3, 
5S = 96 - 3S. 

We cannot see the value of S from this, because we have S's 
on both sides of the equation. But, adding 3S to each side, 
we have: 

6S + 3S == 96 - 3S + 3S 

8S - 96 

S = 12, the smaller number 
and 32 S = 20, the greater number. 

Now let us verify the results;" &c. 



CHAPTER IX 

Vulgar Fractions 

First Notions of Fractions 

Vulgar or Decimal Fractions first? The first notions of 
vulgar fractions will be given in the preparatory Forms, 
where the significance of at least halves and quarters will be 
understood and the manner of writing them down known. 
In- the lower Forms of the senior school, it is probably wise 
first to give a few lessons on the nature and manipulation of 
vulgar fractions, then to proceed with decimals, and to 
return to the more difficult considerations of vulgar fractions 
later. 

The first thing is to get clearly into the child's mind 
that a fraction is a piece of a thing, a piece " broken off " a 
thing. Take one of several similar things (sticks, apples), 
and break off or cut off a " fraction " of it. Cut one of the 
things into 2 equal parts, and introduce the term halves; 
into 3, and introduce the term thirds; into 4, and the term 



68 CRAFTSMANSHIP IN MATHEMATICS 

quarters; and see that the terms halves, thirds, quarters, 
fifths, &c., are made thoroughly familiar. " I have cut this 
apple into 8 parts: give me 1 eighth; give me 5 eighths. I 
add 2 of the eighths and 5 of the eighths together: how many 
eighths have I?" 

We have a special way of writing down fractions. We 
draw a line; under it we write the name of the parts we cut 
the apple into, over it we write a figure to show the number 
of the parts we take: thus 

3 4 



fifths ' sevenths 

Parts of the same name may be added together. Just as we 
say 

2 apples + 3 apples = 5 apples, 

so we may say, 

2 sevenths -}- 3 sevenths = 5 sevenths, 

and we write, 

23 5 



sevenths sevenths sevenths 

Let the child see clearly that the fraction shows 

number of parts 
name of parts ' 

and, a little later on, introduce the terms numerator and 
denominator: 

numerator 
denominator' 

where num = number and nom = name. If the children 
learn Latin, give the Latin words. 

Then come to fractions of collections of things: | of the 
class of children, J a basket of apples, -^ of a Ib. of cherries. 
The way is now paved to fractions of mere numbers: of 32; 
^ of 27; and so on. But at this stage avoid the terms abstract 
and concrete. 



VULGAR FRACTIONS 69 

For illustrating fractional processes, every teacher will 
utilize concrete examples of some kind drawn from everyday 
lifeT "As the number 60 contains numerous^ easy factors, 
fractions of a crown (60rf.) and of an hour (60 minutes) 
make good examples for mental work. 

Mental work may profitably be undertaken as soon as 
the nature of a fraction is fully grasped. 

" Number of pence in | of I/-? in ? in |? in |? in 
yr 2 -? in ^ ?" Let the children thus discover that f = ^ - = -j^-, 
that different fractions may therefore have the same value. 
Thus we come to the notion of " cancelling " and its con- 
verse. 

Again: 

" How many minutes in -3- of an hour? 20. 

i ? 1 r ) 

4 *'' 

1 ? 10 

5 

Thus (5 + J + i) ot an hour 47 minutes. 
How may we express 47 minutes as the fraction of an 

.'. * + i + 2 = U- 



hour? 



Can you see that this is true? No. 

" Well, we have seen that } of an hour = 20 minutes, 
and since 1 minute ---= ( . ] ( j of an hour, 20 minutes = ;? of 
an hour; &c. Hence we may write ^ + \ + -^ in this way: 

eS + IS" + ^ 

and wow it is easy to see why the answer is ^-. 

" Thus if we want to add fractions together, w^jaust 
firstsee that they are fractions^ol the same name, i.e. that 
they have the same denominator. 

" But how are we to change fractions of different denom- 
inators to fractions of the same denominator ?" And sojvve 
cometo L.CJVI.s, &c. 

On" the" whole, however, I prefer to illustrate fractional 



70 



CRAFTSMANSHIP IN MATHEMATICS 



processes by means of diagrams, rectangles rather than lines. 
A rectangle is conveniently divided up into smaller rectangles 
by lines drawn in two directions, and thus the fraction of 
a fraction is easily exhibited. We append a few diagrams. 
(A squared blackboard or squared paper is always advisable.) 

DIAGRAMMATIC ILLUSTRATIONS 
Cancelling. -^ = f = f . 



X* 



Addition.- J + i + | = A + A + A = 



M 



* 



* 



Fig. 2 

Subtraction. f -/ 2 - = -f$. 

A E 




Fig.i 

These figures illustrating subtraction will puzzle the slower 
children, but squared paper and scissors will soon help to 
make things clear. 

Multiplication 

Multiplication by a fraction is always a little puzzling 
at first. A child naturally expects a multiplication sum to 



VULGAR FRACTIONS 



71 



produce an answer bigger than the multiplicand. It is best 
to begin with mixed numbers. 

The child knows that 2/- X 3 is 6/-; by 4, is 8/-; and by 
3, is 7/-. Hence multiplying by the seems to him some- 
how to have been a real multiplication, inasmuch as the 6/- 
has been increased to 7/-. The multiplication may be con- 
sidered, as usual, as an addition, viz. of 3 florins and a half- 
florin. Multiplying a florin by | is to take the half " of " a 
florin. Give other examples to show clearly the meaning of 
the word " of " when we speak of multiplying by a fraction. 

Example: Multiply 2 sq. in. by 3|. 



Fig. 4 

The figure shows 2 sq. in. one above the other, then 
another 2 sq. in., then another, then \ part of 2 sq. in. The 
last piece shows multiplication by a real fraction, viz. 2 in.X^, 
i.e. the strip is \ of 2 sq. in. Thus the whole figure is 6 
sq. in. Hence 2 x 3J = 6|. 

Another example: Draw a figure to show 3J sq. in.\ then 
show this multiplied by 2J. 





S'4 5? In 


9. 










, 




.... 


SyJe*.'. 


.?*-.-.- 











r ' 



Fig. 5 



72 CRAFTSMANSHIP IN MATHEMATICS 

" How from these figures may we obtain an answer to the 
sum 3J x 2|? Let us first think of money, say shillings, 
instead of inches. 

3Js. = 3s. 3d. = 39rf. 

Twice 39<*. = 78^.; one-third of 39J. = I3d. 
/. 2J times 39J. = (78 + 13) J. = 9 Id. = 7*. 7d. = 7^5. 

Apparently, then, 3^ X 2| = 7- 1 1 2 -. Thus, if we divide the 
last figure up into twelfths, we ought to see whether this 
result is true. Fig. 6 shows that it is true; the squares need 
not be true squares; oblongs will show the fractions just as 
well. 



Fig. 6 

Number of twelfths in the figure: 

In the 6 big squares or oblongs, 12 each = 72 

In the 2 strips at the sides, 3 each = 6 

In the 3 strips at the bottom, 4 each = 12 

In the small strip at the corner = 1 

Total 91 

Thus in the figure we have 91 twelfths = f| = 7, as 
expected." 

Such a figure is satisfactory for illustrating the multiplica- 
tion of mixed numbers, but for multiplication purposes 
mixed numbers should rarely be turned into improper 
fractions. A different example is therefore advisable. 

Draw a figure to show f of |. Lead up to the necessary 
figure by showing, first, f; then, by dividing the thirds into 



VULGAR FRACTIONS 



73 





Fig. 7 

7 parts, show y of \\ lastly, show | of f. Obviously, now, 

4 v 2 __ _8 

7 * :i "~ 21"' 

" We have found out that: 

(i), 2 x 3J, or { X 7, - 2 ^; 
and (ii), 3] x 2, or -\ ;J - X J, = *?-; 
and (iii), i X }= / t . 

Now look: in every case the numerators multiplied together 
give the numerator in the answer, and the denominators 
multiplied together give the denominator in the answer." 

Now the teacher is in a position to enunciate the rule. 
He has done nothing to prove the rule, but he has justified 
it, so far as it can be justified with beginners. 

Division 

Just as a child naturally expects a multiplication always 
to produce an increase, so he expects a division always to 
produce a decrease. 

" When you have divided a number by another, the 
dividend is always made smaller. Do you agree?" Yes. 
" Always smaller?" Yes. " Quite certain?" Yes. 

" Let us divide 36 pence amongst some boys. 

I give them 12 d. each: how many boys? 3. 



od. 
2d. 
Id. 



12. 

18. 
ob. 



Do you mean to say I cannot divide the 36 pence 



74 



CRAFTSMANSHIP IN MATHEMATICS 



amongst more than 36 boys?" Yes, if you give them less 
than Id. 

" Then give them %d. each. How many?" 72. 

" Then 36 divided by J is 72. Thus, although I have 
divided 36 I have a quotient bigger than 36. So you were 
wrong!" And so on. 

A suitable scheme of diagrammatic division is easily 
devised, but it is best approached by the division of whole 
numbers. 

When we divide 24 by 4, the quotient is 6, and the 
4 sixes may be arranged in 4 lines thus: 



Fig. 8 



The shaded section denotes the quotient (6 units); it is a 
row of units in line with 1 of the units of the divisor, 4. Any 
other row would have done equally well, for any other row 
would have been in line with 1 of the units of the divisor, 4. 

So generally; a rectangle representing the dividend may 
always be divided up in such a way that each horizontal row 
of units represents the quotient; there are as many horizontal 
rows as there are units in the divisor. Opposite any unit 
of the divisor (we select the first) is a horizontal row of units 
representing the quotient. 

We show 12 divided by 2, by 3, and by 4: 




Fig. 9 



VULGAR FRACTIONS 



75 



Let the first fractional problem be to divide 4 by 1. 
We may ask how many times I^s. is contained in 4s., i.e. 
how many times 16d. is contained in 54d. We may show 
this division in the ordinary way, |-f or - 8 ^, which is equal 
to 3f . Thus the answer to the sum is 3$. How are we to show 
this in a diagram? 

We will first divide 3, represented by a rectangle of 3 sq. in., 
by 4, by 3, by 2, by 2, by 1J, by 1, by f. 



3-r 4= } 




3 - 2J = | 

(Divide AB into 5, and take 

2 parts for the unit.) 







3 -7- 1J = -J 3^1 = 3 

(Divide AB into 3, and take 
2 parts for the unit.) 

Fig. 10 

Examine the six diagrams, and note how the quotient (the 
shaded part) increases as the divisor diminishes. If then 
we diminish the divisor further, the quotient (the shaded 
part) must be still bigger. As before, the shaded quotient 
must occupy a space opposite one complete unit of the 
divisor. But in this case AB is not long enough to show a 



76 CRAFTSMANSHIP IN MATHEMATICS 

complete unit, only |- of one. So we must extend it to make 




3. 4 16 
7f -4- 



Fig. 1 1 



We may now return to our original example: Divide 
by 1J. 




Fig. 12 

One more example: Divide f fry f . 

A 





Fig. 13 

The first figure shows 1 sq. in. 

The second figure shows f sq. in., the part to be 

divided by f . 
The third figure shows AB extended to one complete 

unit, since AB itself represents only f of one. 
The last figure shows the result of the division, viz. 

. 3 8 
~ 4 ^9- 



VULGAR FRACTIONS 



77 



We may now collect up our results: 



5 fl 

2 = 5- 



4 __ 1 5_ 

4 ii 7 

;3 ~~ "s" 

3 _ 

4 1 



An average class will soon discover that by inverting the 
divisor the quotient is then obtained by treating the sum 
as a multiplication sum, e.g. 



But let the teacher be under no delusion. Only a very small 
minority will, at the time, appreciate the purpose of the 
diagrams. Over and over again I have seen a majority com- 
pletely baffled, even with very skilful teaching. No matter. 
Come back to the demonstration again, a year or two later. 
You have justified your rule as far as you can. Now state 
it in clear terms and for the present be satisfied that the 
boys are able to get their sums right. There is probably 
nothing more difficult in the whole range of arithmetic 
than the division of fractions, i.e. for boys to understand 
the process when it is first taught. 

The following kind of argument is sometimes useful: 
To divide a fraction, say 4|, by 5 is the same thing as 
taking of 4|. But to take I " of " 4 is the same thing 
as multiplying 4| by -*-; i.e. 

4J + 5 = 4J x }. 

Now, if we divide 4| by ^, we divide it by a number 7 times 
as small as when we divided by 5; therefore our answer 
must be 7 times as large as before; i.e. 

Since 4J ~- 5 = 4 x -J- 



i.e to divide by |- is the same thing as multiplying by -J. 
Hence, once more, the rule of inverting the divisor. 

But the argument is quite beyond the average beginner, 
as every experienced teacher knows. 



78 CRAFTSMANSHIP IN MATHEMATICS 

Cancelling. When a boy is told he may " cancel 
thus: 

2 5 



he is likely to ask, why? He will already have learnt: 

(1) That, e.g., 7 X 9 - 9 X 7; 

(2) Reduction of fractions to their lowest terms; 

(3) Multiplication of fractions. 

Thus he will understand that 

If x = 44 x 25 

46 22 ~~ 45 X 22 ' 

and that this may be written: 

44 X 25 44 25 

_ _ __ QJi _ ^ _ 

22 x 45 22 45' 

He now readily sees that he is justified in reducing each 
of these to its lowest terms, and that the final result is the 
same as when he cancelled terms in different fractions. 



CHAPTER X 

* t 

Decimal Fractions 

A Natural Extension of Ordinary Notation 

If care is taken to teach the inner nature of decimal 
notation thoroughly, decimals need present little difficulty. 

" A decimal fraction is merely a particutyr kind of vulgar 
fraction, viz. one with a denominator 10 or power of 10, 



DECIMAL FRACTIONS 79 

e -g- Tb- T<io> Hoo> Toto o- But we do not generally write 
them this way; we write them as follows: 

7, -73, 8-192, -0003 

" Let us consider a number consisting entirely of ones 
(any other figure would do equally well). 

The number 11 = 10 + 1 

The number 111 = 100 + 10 + 1 

The number 1111 - 1000 + 100 +10 + 1. 

The 1 of least importance in each number is the 1 on the 
extreme right; each 1 to the left is 10 times as important 
as, and is 10 times the value of, its right-hand neighbour. 
The 1 on the extreme right represents just one unit. 

" But we often break up a unit into parts, e.g. a sovereign, 
or a bag of nuts. These parts are fractions, and we might 
carry our ones to the right, to represent these fractions, 
devising some means of separating the fractions from the 
whole numbers: a straight line would do. 

Thus, 1111|1 may stand for 1000 + 100 + 10 + 1| + ^ 
and, 11|111 may stand for 10 + 1|+ ^ + t ^ + 



Any sort of separating mark will do. Generally we use the 
smallest possible mark, a dot, written half-way up the height 
of the figures. 

" Thus 276-347 - 200 + 70 + 6 + -^ + jfo + T ^. 
If we add the 3 fractions together, we get -nrcnr Thus 



276-347 ~ 276 v 

" Note the two ways of writing down the same thing: 

~ 



_ 4 _L_ 5 i i 1 _ 4501 

i o ~r TOCT ~r Twer ~r ~ 



__ _ 

i o ~r TOCT ~r Twer ~r unnnr ~ ioooo 

" Perhaps there are no tenths, and we begin with 
hundredths: 

042 = 



We could not write this -42 because -42 Y 



80 



CRAFTSMANSHIP IN MATHEMATICS 



Thus we may have one or more noughts between the decimal 
point and the " significant " figures. 

" But noughts on the extreme right hand of a decimal 
have no meaning: 

034 = A + T? 

1 ' 1 i 

as before/' 

Give ample practice in conversion and reconversion, until 
the change can be written down mechanically: 

37-063 == 37^^; ^g^ r- -0421; &c., &c. 
'conversion of either decimal form into the other is essential. 



4 JJ4 

iodo 1000 

T(JCT(J + 



o _ 34 

1(7(50(1(1 ~ fooo 



This is tfie key to a 



Let the rules for conversion be stated in the simplest 

possible words. 

A few exercises of the following nature are useful: 

" If x 10, find the value of these expressions, writing 

down the answers in both decimal forms (do not cancel as 

you would vulgar fractions): 

3 + - + - ? 2 = 34- , 4 + Jff = 3 t Vo, or 3-47. 
x x 2 

6r> + 4* + 9 -j- ~ + ~ = 6049 1 g-gg , or 5049-0306. 



Multiplication and Division by 10, by 100, &c. 

" If we multiply 347 by 10 we obtain 3470, the 7 becoming 
70, the 40 becoming 400, and the 300, 3000. 



Th. 


H. 


T. 


u. 




3 


4 


7 


3 


4 


7 






Every figure is moved one place to the left, and its value is 
increased 10 times. 



DECIMAL FRACTIONS 



81 



" If we divide 2180 by 10 we obtain 218, the 2000 be- 
coming 200, the 100 becoming 10, and the 80, 8. 



Th. 


H. 


T. 


u. 


2 


1 


8 







2 


1 


8 



Every figure is moved one place to the right, and its value 
is diminished 10 times. 

" So it is with decimal fractions, or decimals as we often 
call them." 



Multiply 3*164 by 10. 

3-164 x 10 = (3 + t V + T n + 
= 30+1-1- ;' + 1( 
=-- 31 f Vo -31-64; 



X 10 



i.e. the decimal point has been moved one place to the right, 
and every figure occupies a place 10 times as important as 
before. 

So 5-623 X 100 = 562-3; 

005623 x 10000 = 56-23. 



+ 



Divide 3-164 by 10. 

3-164 X = (3+ ,',-, + , 
= > 3 <> + i w + 

'M (14 _ 

1 O (7 

i.e. the decimal point has been moved one place to the left, 
and every figure occupies a place reduced in importance 
10 times. 

Give a number of varied examples in both multiplication 
and division, and help the pupils to deduce the rules. 

Give plenty of mental work of the following kind: 

tens X tens = hundreds 

hundreds X tens = thousands 

hundreds X hundreds == ten thousands 
tenths X tenths = hundredths 

hundred ths X tenths = thousandths. 
(B 291 ) 7 



82 CRAFTSMANSHIP IN MATHEMATICS 

Continue this kind of work until instant reponse is obtained 
as to the significance of moving the decimal point so many 
places to the right or so many to the left. Let the notation 
be mastered; then the rest will give little trouble. 

Addition and Subtraction 

Do not forget the common cause of inaccuracy, blanks 
in the fractional columns, especially if the numbers are 
arranged horizontally, e.g. 7-612 + 3-1 -f 2-0151. 

Multiplication 

72-314 X -32 

_ 79 31 4 vy 32 

~~ 



__ 7 'J 3 1 4 v 3 2 
-- TT700 X lOtf 
_ 2 3 1 4 t H 
-- 10 (TO GO 

= 23-14048. 

The whole process resolves itself into (1) conversion, (2) 
multiplication of whole numbers, (3) reconversion. 

Note that " conversion " does not mean conversion to 
vulgar fractions, but to the alternative form of decimal 
fractions, with denominators consisting of powers of 10. 

The multiplication of the denominator is really nothing 
more than the mere addition of noughts, and it is obvious 
from this multiplication that the number of decimal places 
in the product is equal to the sum of the numbers of decimal 
places in the multiplicand and multiplier, and that from the 
very nature of the case this must always be so. 

Hence the simple rules: 

1. Ignore the decimal point and perform the multiplication 
as if the multiplicand and multiplier were whole numbers. 

2. Add together the decimal place in the multiplicand and 
multiplier; this gives the number in the product. Fix the 
point by counting back that number of places from the 
right. 



DECIMAL FRACTIONS 83 

72-314 
32 

144628 
21 6942 
23-14048" No. of dec. places = (3 -f 2) = 5 

Is the method intelligent^ It is at least as intelligent as 
any other method, and it has this advantage that the boy 
works his sums exactly as he works ordinary simple mul- 
tiplication. And the procedure is easily and immediately 
justified, by conversion and reconversion. 

Give other examples, using the same numbers but changing 
the position of the decimal points. The answers shall be 
given mentally and at once: 

72314 X 3-2 = 2-314048. 
0072314 X 32 = 0-2314048. 
72314 X -00032 = 23-14048. 
723-14 X 320 = 7231-4 X 32 
= 231404-8 

If preferred, the boy might set out his working thus: 
72-314 x -32 = ^4 x ^ 

2 31 4 4 R 

~iTyfrofy<r 
= 23-14048 

and show his actual multiplication neatly on the left. 
Possible objections to the method. 

1. " The most important digit in the multiplier is not 
used first." Granted. But this disadvantage is outweighed 
by the advantage of greater accuracy. 

2. " The decimal points are not kept in a vertical column." 
This is of no material consequence, though it is quite easy 
to teach the boy, if it is thought worth while, to place the 
points in the successive products. For instance: 

72-314 

-32 

1-44628 
21-6942 
23-14048 



84 CRAFTSMANSHIP IN MATHEMATICS 

The boy multiplies through by 2 and then says, " When I 
multiplied 4 by 2, I multiplied thousandths by hundredths, 
and this gives me hundredths of thousandths, which occupy 
the 5th decimal place; therefore the point goes in front of 
the first 4." He argues similarly when he has multiplied by 
the 3, though he would soon learn that the position of the 
point in the first partial product gives the key to its position 
in all the other products. Thenceforth he would work 
mechanically. Does not the time come when we all work 
mechanically in all types of calculation? does not the rationale 
of procedure tend to fade away, until something turns up 
demanding revivification? 

Is there a more intelligent plan than teaching the boy to 
complete the actual multiplying before considering the 
decimal point at all? I doubt it. And I am quite sure that 
no other plan is productive of a greater degree of accuracy. 
The boy has confidence in a method so closely akin to one 
with which he is already familiar. 

Standard Form 

It has been gravely said that " standard form " was the 
invention of the devil. In reality it was not quite so bad as 
that. It was invented* by an old personal friend of my own, 
the senior mathematical master of one of our great Public 
Schools, who decided that he " must adopt some new method 
to prevent his boys from getting so many sums right, in order 
to take the conceit out of them ". 

Why are the apologists of the method always so faint- 
hearted? 

* The method was suggested by '* standard " form in logarithms, where of 
course it is very useful. 



DECIMAL FRACTIONS 85 



Division 

As division is the reverse process of multiplication, the 
analogous method for fixing the decimal point may be adopted. 

23-14048 -f- -32 

_ 2 :i 1 4 O 4 N 





_ 2 .'I 1 4 4 H 

---- a a 



= 72314 x 

__ 7 2 :) 1 4 
-- 1 006" 

= 72-314. 



The actual simple division by 32 may be neatly shown to the 
left. 

The simple rules are: 

1. Ignore the decimal points and divide as in simple 
division. 

2. Subtract the number of decimal places in the divisor 
from the number in the dividend. This gives the number 
in the quotient. Fix the point by counting back this number 
from the right. 

The whole process resolves itself into (1) conversion, 
(2) division by whole numbers, (3) reconversion. 

Again: is the method intelligent? Again the answer is that 
it is as least as intelligent as any other method, and it certainly 
makes for accuracy. Here is an example with the working 
as commonly shown: Divide 2-0735 by 8-72. 

_ 23 
8-72)2-0735 
1 744 

~3295 Decimal places = 4 2 = 2. 
2616 Thus the quotient is "23, and a remainder. 
1379 

The division may now be continued to any number of places. 

If, before dividing, we add O's to the dividend and 

continue the dividing further, this does not affect the 



86 CRAFTSMANSHIP IN MATHEMATICS 

decimal point in the quotient: e.g. divide 2-073500 by 
8-72. 

2377 

8-72)2 : 073500 
1 744 
3295 
2616 



6790 Decimal places =6-2 = 4. 

6104 Thus the quotient is '2377, and a remainder. 

~6860 

6104 

~756 

Hence, if a given dividend contains a smaller number of 
decimal places than the divisor, add O's to make the number 
equal (and more if necessary). Example: divide -001 by 

7-0564. Write: 

7-0564>OOl 

We cannot proceed with the division until we add at least 
5 more O's. 

1 
7-0564y00100000 

70564 Decimal places = 8-4=4 
29436 

Thus the answer is '0001 . . . and a remainder. The quotient 
can be carried to as many places as may be required. 

The value of the remainder. It is desirable to make 
the abler boys see the real value of the quantities in the 
successive steps of the division. Example: divide '07925 
by 3-7. 

214 

3-7)-07925 
74 
62 

37 Decimal places = 51 = 4. 
155 Quotient = '0214 and a remainder. 
148 
7 



DECIMAL FRACTIONS 87 

What is the value of the " 74 " in the first step? It is the 
product of 3-7 and -02 (as we may see from the quotient), 
and must therefore contain (1 + 2 or) 3 decimal places; 
hence its value is '074. Similar arguments apply to each 
step. Hence, more correctly, the division may be set out in 
this way: 

0214 
3-7JM37925 

074 = 3-7 X -02 

0052 

0037 = 3-7 X -001 

00155 

00148 = 3-7 X -0004 



00007 

Thus the quotient (to 4 figures) is '0214, and the remainder 
is *00007. The abler boys will soon learn to assign the 
correct value to the remainder, by merely glancing at the 
dividend vertically above it. 

Verification should be encouraged: 

Dividend = (quotient X divisor) + remainder 
= (-0214 X 3-7) + -00007 
= -07918 + -00007 
= -07925 

Practice in manipulation of the following kind is useful: 

3-204 = 3204 = -03204 = 320-4 
^0701 ~~ ~7(M ~ -000701 ~ 7-01 ' 

The boy sees at once that the same quotient must result 
from all the division sums. The only real defence for the 
reduction of the divisor to a form approximating a small 
whole number is that it enables a boy to obtain a rough 
answer by easy calculation. For instance, in the last of the 

r r 1. 3204 320 , . , , 

four forms above, -=-^r- = -=~ approximately, and thus the 



88 CRAFTSMANSHIP IN MATHEMATICS 

answer to this group of division sums is roughly y of 320, 
i.e. a number between 40 and 50. 

. '. -000983 -0983 -983 

Again: -^- = or _ . 

Thus the answer is roughly ~ 4 ~ 7 - of 98 hundredths, i.e. about 
2 hundredths, i.e. about -02. 

This is useful for final verification, but the decimal point 
is best fixed by the rule already given. If the simple multi- 
plication and division are accurately performed, the correct 
fixing of the decimal point is a simple matter to even an 
unintelligent boy. 

Recurring Decimals 

These will probably rarely be used, except in a very 
simple form. Every boy ought, however, to know their 
significance, though as a subject of general exposition they 
are now generally ignored, perhaps unwisely. The younger 
race of mathematicians are losing familiarity with much 
that is interesting in the theory of numbers. Most people 
know of course that \ - -142857142857 . . .= -14285?, and 
that if we multiply this group of 6 figures by 2, 3, 4, 5, and 
6, respectively, we obtain products giving the same group 
of figures in the same order, each succeeding group be- 
ginning with the next higher figure of the group. It is, 
however, less commonly known nowadays that this re- 
markable property of numbers is not uniquely characteristic 
of the sevenths but applies to all prime numbers whatsoever, 
7 and beyond, and that the grouping within the groups is 
sometimes of an extraordinarily interesting character. Teachers 
of arithmetic probably lose not a little of the potential effective- 
ness and interest in this subject if they do not familiarize 
their pupils with some of the properties of numbers, properties 
which to beginners seem almost uncanny. (See Chapter XLI.) 

As to circulating decimals, a boy should be taught at 
least this much: 



DECIMAL FRACTIONS 89 

Show him that he may at any point bring to an end the 
quotient of a decimal he is dividing, showing the remainder 
as a vulgar fraction. 

Thus I = -14; or -1428 J or -142852. 

Reconvert, say, the first: -14= - ^ + ^ = 1^1 = '<><> = *. 

Then the boy sees that the scheme is justified. 

Again: -J = 3333 . . . apparently without end, 

so, J = -1111 . . . apparently without end, 

and, J,J -717171 . . . apparently without end. 

But we can bring the division to an end anywhere, e.g.: 

i = -3333J, or -33J or -3. 
1 = -111?, or-lj,. 
H =-7171JJ or-71JJ. 

Reconvert these: 

ol 

.01 _ ; __ 1 O __ :i O _ 3 

;i "" io~" ;l ~ '*-- 

1 1 

.11 . 1< i i _ 10 _ 1 

'' ~ ~ lltf ~ "' 



10 

.71?! _ 7 _i x i) it _ 7 i 170 _ 7100 _ 71 

' - 1 '. ( .) 10 ' TA i i" u y o i> o o '.) TT 



171 

x i) it _ 



Now show the repeated figures in a decimal division this way: 
3333... = -3; -1111... = -1; -717171... = -tl. 

Thus we have learnt that 



Hence to convert any repeating decimal into a vulgar fraction, 
we make a denominator of 9's, viz. as many as there are places 
in the decimal. 

Thus -Si = fJ; '69 = = 



90 CRAFTSMANSHIP IN MATHEMATICS 

If there are non-repeating figures as well as repeating figures, 
e.g. -5l6, then 

1A 6-ia_6j!!-_ 811 . 
.516 = _ - _ - , 

.01 A- l ' 6 - l " - i - i 

016 - loo " 100 ~ TW ~ **' 

The commoner forms should be known, especially the thirds, 
sixths, and twelfths: 



I = .16; = -83. 
^ = -083; ^ = -416; -/^ = -583; }4 = -916. 

The boys should know that when the denominator of a 
vulgar fraction contains any prime numbers except 2 and 5, 
the conversion of the fraction to a decimal is bound to give 
a circulating decimal. 

Simplification of Vulgar and Decimal Fractions 

There are certain conventional rules about signs; for 
instance: 

1. Multiplication and division must be given precedence 
over addition and subtraction: 

3X 18 +15 -f-3 2 means (3 X 18) + (15 ~ 3) - 2. 

2. Multiplication and division alone must be worked in the 
natural sequence from left to right: 

36 -r 9 X 2 means (36 ~ 9) X 2. 

But the conventions are not whole-heartedly accepted; they 
are without reason, and they are traps for the unwary. It is 
unjust for examiners to assume that they will be followed. 
The above examples should have been written with the 
brackets. If brackets are inserted mistakes need not arise. 

Here is a complex fraction to be simplified, taken from 
one of the best textbooks in use. Doubtless the question 



DECIMAL FRACTIONS 91 

was taken from an examination paper. If so, the examiner 
should have been put in the stocks. 

1-463 jj of 141-75 - j X 88-125 
7-315 5 X 18-9 + 25 + 1-22 

If given at all, it should have been written 
^ 3 + f (lon^M^^ 

7-315 I (5 X 18-9) + 25 -f 1-22 






It is often an advantage to work in decimals instead of 
vulgar fractions. Example: What fraction of 21, 5$. 6d. 

. -04255 X -32 f ri , , 
1S . -- of /I, 11$. 3rf.? 
00016 * ' 



. -00016 

rraction = 



04255 x -32 -,. OK ,. |f 
of 31-25 shill. 



425-5 shill. 
04255 x -32 x 31-25 
^00016 X 425-5 

1 2 

x & x 3125 



x 

1 1000 

= 6-25 



At the third step, both numerator and denominator were 
multiplied by 10 9 , to get rid of the decimals. Boys feel more 
confidence when cancelling whole numbers. But in A Sets 
such conversion should not be necessary. 

Decimalization of Money 

The common method of performing arithmetical opera- 
tions on money, weights, and measures reduced to their lowest 
denomination has the advantage of simplicity but the dis- 
advantage of tediousness and cumbrousness. It is certainly 
an advantage to work in the highest denomination when 
possible, decimalizing all the lower denominations. For 



92 CRAFTSMANSHIP IN MATHEMATICS 

instance, if we have to multiply 432, 175. \d. by 562, it is 
obviously an advantage to multiply 432-86875 instead, 
provided we can convert into the decimal form at once. 

But the rules for conversion to more than 3 places are a 
little too difficult for slower boys, and it must be remembered 
that if multiplication is in question (and this is often the 
case), exact decimalization is necessary, or the multiplied 
error may be too serious to be negligible. The mil invariably 
causes trouble. On the whole, decimalization methods are 
advisable in A Sets, not in others. 

But all boys should be taught to give in pounds the 
approximately equivalent decimals of sums of money, i.e. 
to call every 2/-, !; every odd I/-, -05; every -/6, -025; 
every farthing, -001. 

Thus: 3, 17s. lOJrf. 

- 3 + 16/- + I/- + -/G + IS/. 
= (3 -f- -8 + '05 -f -025 -|- -018) 
= 3-893. 

Greater certainty and greater accuracy is obtained by the 
ordinary method: 

\d. - -&d. 



= -875.9. 

175. lOJrf. - 17-8755. 
= -89375 
3, 17s. 10JJ. - 3-89375, 

the boys dividing by 12 and 20 without putting down the 
factors. 

The converse operation, the conversion of decimally 
expressed money into pounds, shillings, and pence, is most 
safely and quickly performed by the old-fashioned multiplica- 
tion method (by 20 and by 12). 

Consider 307-89275. 

-89275 = 17-855s. 
8555. = 10-26rf. 
.'. 307-89275 == 307, 175. 10 26rf. 

= 307, 17s. 10^. (+ -Old.). 



DECIMAL FRACTIONS 93 

Numerous tests in recent years have shown conclusively 
that the usual decimalization rules are productive of much 
inaccuracy amongst slower boys. But decimalization and 
reconversion by the ordinary methods of division and multi- 
plication are easy to effect and are often advantageous in 
practice. 

A quick boy who wanted to multiply 15s. 9rf. by 2420 
would probably use the practice method (15s. = of 1, 
9rf. = ~^ ( - of 15s., and he would see at once that the product 
is 1815 + 90-75 or 1905, 15s.), and he would not 
decimalize. But to a slow boy a choice of methods is only 
an embarrassment. He wants one method, and that method 
without frills of any kind. 

Contracted Methods 

Contracted methods of multiplying decimals are productive 
of so much inaccuracy that their use with average boys is not 
advised. In A Sets, of course; in B Sets, perhaps; in C and 
D Sets, no; though in A and B Sets logarithms will usually 
be used instead, unless the sum to be worked is so simple 
that ordinary methods are quicker. To slower boys logarithms 
are puzzling, and their use in lower Sets is not recommended. 
But no boy ought to be allowed, in Forms above the Fourth, 
to show in his working the figures to the right of and below 
the heavy line in a sum like the following. 

Divide 5-286143 by 37-29 (to 4 places). 

-1417 

3X^)5-286143 
3729 
1557 1 
1491 6 



65 
37 

82 



54 
29^ 
253 



26 103 
2 150 



94 CRAFTSMANSHIP IN MATHEMATICS 

I have found that even slower boys soon gain confidence in 
cutting out figure after figure in the divisor, instead of 
bringing down figures from the dividend. The doubtful 
" carry " figures worry him a little at first, but not for long, 
and he soon learns to understand what to do to ensure 
accuracy to a given number of decimal places. 

Never encourage average boys to adopt the expert mathe- 
matician's plan of multiplying and subtracting at the same 
time (Italian method). Boys hate it, rarely become expert 
at it, and make mistakes galore. It should of course be used 
by boys having any sort of real mathematical bent. 



CHAPTER XI 

Powers and Roots. The A, B, C of 
Logarithms 

Powers and Roots 

I have seen four- and five-figure logarithms deftly used in 
Preparatory Schools, but it is probably not wise to expect 
much facility before the age of 14. The A, B, C of logarithms, 
as a simple extension of work on powers and roots, may, 
however, readily be taught a little sooner. 

At first, powers, indices, and roots should always be 
treated arithmetically, not algebraically. The later general- 
izations are then much more likely to be understood. 

Some typical preliminary exercises: 

(i) 5 4 = 5 X 5 X 5 X 5; 5 3 = 5 X 5 X 5; 
.-. 6* X 6 3 = (5 X 5 X 5 X 5) X (5 X 5 X 5) 

= (5x5x5x5x5x6x5) 

= 5% 
/. 6 x 5* = 5 7 . 



POWERS, ROOTS, LOGARITHMS 95 

Thus lead up to the rule, and then state it clearly, that 
in multiplication of this kind the indices are added. But 
impress on the boys that the operation concerns powers of 
the same number (in this case, 5), though any number may be 
similarly treated. 

(ii) 7 9 = 7x7x7x7x7x7x7x7x7; 

7 4 = 7x7x7x7; 



\X\X\X\ 



79 

L = 7-4 = 75 

74 



Now lead up to the associated rule of subtraction of indices, 
in division of this kind. 

(iii) 7 4 = 7X7X7X7; 

7 4 _7x7x7x7__, 

" 7 4 "" 7~X 7 X 7~x~7 ~ ' 

. 74-4 ^ i or 70 = i So 30 ^ 1; 10 o = j^ 

(iv) (7 2 ) 3 - 7 2 X 7 2 X 7 2 

- (7 X 7 X 7 X 7 X 7 X 7) = 7 6 . 

Thus lead up to the rule as to multiplication of indices. 
State categorically that such results always hold good, and 
that a convenient way of remembering them is this: 



But at this stage do not talk about " general laws ". Let 
the above expressions be looked on merely as a kind of short- 
hand for collecting up several results actually worked out 
arithmetically. 

The following is a summary of a particularly effective 
first lesson I once heard, given to a class of boys of 13, on 
fractional and negative indices. 

" The square root of a number is that number which 
when squared produces the original number; e.g. the square 
root of 16 is 4; of 81 is 9. We write R16 = 4; R81 = 9. 



96 CRAFTSMANSHIP IN MATHEMATICS 

(The mathematician writes his R like this: ' v /> an d calls 
it ' root '. Thus * V 36 = 6 ' reads ' root 36 is 6 ') 
carefully that V 6 X V 6 = V^ = V 62 = 6 ; that 
X y'll = 11; and so on. 

" The cube root of a number is that number which when 
cubed produces the original number. We show the operation 
by writing a little 3 inside the <\A Thus 4/125 = 4/5 X 5 X 5 
- 4/5 3 - 5; 4/1000 - 4/10 X 10 X 10~= 4/10 3 - 10; 
4/5 X 4/5 X 4/5 = 4/12.5 = 5. _ 

" And so on. 4/81 - 4/3~X 3x3x3 - 4/3 1 = 3. 

" Now suppose the index is a fraction, and not a whole 
number. What does 5* mean? Well, we have learnt that 
52 x 5 2 = 5 2 + 2 = 5 4 , so apparently we may assume that 
5* X 5* - 5*+* = 5 1 - 5. 

" But V 5 X V 5 = 5; therefore 5* = V 5 - 
In other words, 5* is merely another way of writing down \/5. 
Similarly, 5* X 5* X 5* - 5* + * f * - 5 1 - 5; therefore 
5* = 4/5, i.e. 5* is another way of writing down 4/5. 

" Similarly, 4/7 - 7 1 ; 4/2 - 2*. 

" What does 8* mean? We know that (3 4 ) 3 = 3 12 , so 
apparently we may assume that 8* = (8 2 )*, i.e. that 8 J means 
the cube root of 8 2 , or 4/8 2 , or 4/64, or 4. 

" Similarly 5 3 = 4/5 3 - 4/125. 

" Thus we have learnt that the numerator of a fractional 
index indicates a power ', and that the denominator indicates 
a roo. 

" Again, what does 6~ 2 mean? 

" Since 6 5 x 6 2 = 6 5+2 = 6 7 , apparently we may assume 

that 6 5 X 6~ 2 - 6*- 2 - 6 3 . 

But & -f- 6 2 = ^ - 6 3 ; 

.'. 6^ X 6~ 2 - 



or 



POWERS, ROOTS, LOGARITHMS 97 

Similarly, 5- 3 = -I ; also 1 = 7~ 4 . 

Thus we may conveniently remember that a~ n and are 
two ways of writing down the same thing. 

" Examples: 



7 -i _ _ 

75 



" 



A root form in a denominator is often troublesome, 
since it leads to difficult arithmetic; and we may often get 
rid of it in this way: 

1= J_ X ^=^= W6" 

A/5 V5 V5 5 " 

The lesson was followed up by a few very easy exercises. 
There was no algebra, save the " shorthand " expressions 
utilized as mere mnemonics. The teacher's purpose was to 
make the boys familiar with the basic facts of indices (integral 
and fractional, + and ), and with the alternative forms of 
writing down the same thing. Naturally many more examples 
were given than the few above cited, and by the end of the 
lesson the boys were remarkably accurate in their answers to 
" mental " test exercises that were made somewhat severely 
searching. 

Boys should know their squares up to 20 2 . Extraction of 
square roots may be taught when (a + b) 2 is known in algebra, 
though boys should be made to break up numbers into 
factors whenever possible, and then to obtain square roots 
by inspection. Encourage boys to leave certain types of 
answers in surd form, but, generally, to rationalize their 

denominators; thus the answer - would not be acceptable, 

V 7 

(E291) 8 



98 



CRAFTSMANSHIP IN MATHEMATICS 



but y 5\/7 would. All boys should know the values of 
V'S, -\/5, \/T, to 2 places of decimals. 

The Beginnings of Logarithms 

" Mathematicians were long ago clever enough to see 
how they could use indices for working long sums in multi- 
plication and division. Suppose they wanted to multiply 
together two large numbers, each of which was a power of 3. 
They would turn to a book of * tables ' showing the powers 
of 3. In fact, we may easily make up a little table for ourselves: 
e.g. 3 l - 3; 3 2 = 9; 3 3 = 27; 3 4 - 81; &c. Here is a 
table from 3 1 to 3 16 . In the first column we write the index, 
in the second the corresponding number. 



Index. 


Number. 


Index. 


Number. 


1 


1 


9 


19,683 


2 


9 


10 


59,049 


3 


27 


11 


177,147 


4 


81 


12 


531,441 


5 


243 


13 


1,594,323 


6 


729 


14 


4,782,969 


7 


2187 


15 


14,348,907 


8 


6561 


16 


43,046,721 



" Now for some exercises. 

1. Multiply 19,683 by 729. 

Answer: 19,683 x 729 = 3 9 X 3 6 = 3 16 = 14,348,907. 



2. Divide 43,046,721 by 531,441. 
Answer: 



531,441 



3 12 



3. What is the square of 2187? 

Answer: (2187) 2 - (3 7 ) 2 = 3 14 



= 81. 



4,782,969. 



How easy! Instead of working hard sums, we simply refer 
to our table, and add, subtract, or multiply little numbers like 
9, 6, &c. 



POWERS, ROOTS, LOGARITHMS 99 

" But the mathematician would set out the first sum some- 
thing like this: 

Index of the answer = index of (19,683 x 729). 

= index of 19,683 + index of 729. 
= 9 + 6. 
= 15. 

" But in the table the number corresponding to the index 
15 is 14,348,907. 

.'. Answer = 14,348,907. 

Rather a roundabout way, isn't it? And he has a rather 
grand word which he prefers to the word index, it is ' log- 
arithm '. The second part of the word, -arithm, you already 
know; the first part, log means * rule ' or ' plan '. Although 
logarithms are only indices, the word itself suggests a clever 
" arithmetical plan " for shortening our work, and you must 
try to master it. 

" Our little table contains only a few numbers, and there 
are big gaps between them; e.g. there is no number between 
27 and 81. Now 27 - 3 3 and 81 = 3 4 . Would it be possible 
to obtain a number between 27 and 81 by finding the value 
of 3 3 *? Certainly we should suspect that the value of 3 3 * 
is somewhere between 27 and 81. 

" We know that 3 3 * - 3* - ^ = ^2187 = 46-8 (by cal- 
culation). Hence 3 3 * does lie between 3 3 (27) and 3 4 (=81). 
Obviously it is possible to put into our table as many fractional 
indices as we like, and so make the table more complete. 

" The 3 which we have made the base of our calculations 
the mathematician calls a base. Any other number might 
be used instead, and in point of fact 10 is generally used." 

The boy is now in a position to understand that (base) 100 
= natural number, and he may be introduced to a short 
table of three-figure logarithms, a table that may be included 
in a single printed page. Give a variety of very easy examples, 
and avoid great masses of figures. It is enough at this stage 
to drive home the main principle. There is much to be said 



100 CRAFTSMANSHIP IN MATHEMATICS 

at first for avoiding the word logarithm altogether, and for 
letting the boy work from the relation, number = lO indMf . 
But we are anticipating Form IV work. (See Chap. XVII.) 



CHAPTER XII 

Ratio and Proportion 

Simple Equations Again 

If 4 chairs cost 20, what is the cost of 15 chairs? 

4 chairs cost 20. 

/20 

.*. 1 chair costs . 
4 

^ i 20 X 15 --.., 

/. 15 chairs cost ^ 75. 

4 

This method, " the method of unity ", is a good childish 
way of working such a sum, and it is the method suitable 
for boys up to the age of 11. At about this age the notion of 
ratio should be introduced and it should gradually supersede 
the unitary method. 

First, revise the work on very simple equations. 

We may begin a sum by saying, <k What number of 
chairs . . . ?" The number we do not know; we have to 
discover it. It is customary to let the symbol x represent a 
number not yet discovered, and to argue about the x just as 
we argue about any ordinary number. 

How many chairs can I buy for 45, if 1 costs 

Let the number of chairs be x. 

Then 5 x x = 45, 
/. 5* = 45. 
/. * = Y = 9- 

Similarly, if 11 x = 51, x = || 3. 



RATIO AND PROPORTION 101 

" We have already learnt that we may multiply or divide 
the two sides of an equation by any number we please, 
provided that we treat the two sides alike; e.g. 

If x = 12, then 3x = 36, or ~ = and so on. 

If we have an equation involving fractions, it is an advantage 
to get rid of them as soon as we can, and we may always do 
this by multiplying both sides of the equation by the L.C.M. 

x 20 

of the denominators: e.g. let the equation be - = . The 

L.C.M. of 9 and 12 is 36. Multiplying both sides by 36, 
we have 

4x --= 60. 
/. x = 15. 

Instead of using the L.C.M. for our multiplier, any other 
C.M. will do, though this will mean rather harder arithmetic. 
We might, for instance, use the product of the denominators, 
viz. 108. 

x = 20 

9 12* 
/. 12jc = 9 X 20. 

x 15 (as before). 

In this form we see in the simplified second line all four 
terms of the original equation (x, 9, 20, 12), and this sim- 
plified second line might have been obtained at once from 
the original equation by cross multiplying, i.e. by mul- 
tiplying each numerator by the opposite denominator. 
This cross-multiplying is often very useful, in algebra and 
geometry as well as in arithmetic. 

" From cross-multiplying it follows that if we have two 
equated fractions, a numerator and the opposite denominator 
may be interchanged; e.g. = \ -* , 2 7 8 = J .f , -^ 2 ~ = - 2 \. 

" Now we come to Ratio and Proportion." 



102 CRAFTSMANSHIP IN MATHEMATICS 

Ratio and Proportion 

If 1 sheep cost 3, then, 



2 sheep cost 6 

3 9 

4 12 

5 16 



7 sheep cost 21 
10 30 
13 39 
21 63 



As we increase the number of sheep we increase in the same 
proportion the number of pounds. 

Take any pair of numbers (sheep) from the first column, 
and the corresponding pair of numbers (pounds) from the 
second, say the last but two and the last in each case, and 
convert them into fractions, thus: 



153* 



We see that these fractions are equal. That we should 
expect, for 10 bears the same relation to 21 as 30 bears 
to 63. A better way of saying it is that the ratio of 10 to 21 
is equal to the ratio of 30 to 63. 

" We know that the sign of division is ~, and that if in 
the place of the two dots we write numbers, e.g. |, we have 
a fraction, and that the fraction means 5 divided by 6. Thus 
a fraction represents a quotient. Similarly a ratio represents 
a quotient. A ratio merely shows the relation between two 
quantities, viz. how many times one is contained in the other. 
When two ratios are equal, as in the case of the sheep and 
pounds, we write them thus: 

10 30 

aT *3> 

and we read, 

10 bears the same ratio to 21 as 30 bears to 63. 

Such a statement is a statement in proportion. Sometimes 
we read ' 10 is to 21 as 30 is to 63 ', and sometimes * 10 over 
21 equals 30 over 63 '. 

" Remember, then: a statement in proportion is a 
statement of the equality of two ratios. 



RATIO AND PROPORTION 103 

How many pounds of tea can I buy for 40$. if 6 Ib. cost 15s.? 
Call the unknown number of pounds, x. We have 4: terms 
viz. 2 lots of pounds, 2 lots of shillings. Write: 

Lb. Shillings. 
x cost 40. 
6 15. 

Convert each pair of terms into a ratio or fraction, equate, and 



6 15" 

.'. 5x = 80. 
.'. x - 16. 

16 Ib. of tea cost 40s.; how many pounds can I buy for 155.? 

Lb. Shillings. 
16 cost 40. 
x 15. 

16 40 

Equating, = . 

x 15 

.-. 40# = 240. 
/. x = 6. 

Find the cost of 6 Ib. of tea if 16 Ib. cost 405. 

Lb. Shillings. 
6 cost x. 

16 40. 

Equating, 1 = i. 

/. 2x = 30. 
.-. x - 15. 

6 /6. of tea cost 15$.; etfAtf* is the cost of 16 Ib? 

Lb. Shillings. 
6 cost 15. 

16 x. 

v 6 15 

Equating, --- = . 

16 x 

:. Qx = 240. 



104 CRAFTSMANSHIP IN MATHEMATICS 

The simple scheme applies to all cases of direct proportion 
write down the 4 terms in pairs; equate; solve for x. 

I sell a horse for 47, 10$., thereby losing 5 per cent. What 
should I have sold him for if I had gained 5 per cent? 

n . . ^ , Representative 

Prices tn pounds percentages. 

47J 95 

x 105 



47J 95 
Equating, - = 



= 52, 



95 2 

The prices of. the horse are in direct proportion to the re- 
presentative percentage numbers. 

Inverse Proportion 

But, of course, inverse proportion is another story. In 
practice it is relatively rare, and is thus sometimes overlooked. 

To cover a floor with carpet 72 in. wide I require 40 
yd. from the roll; if the carpet is only half the width, I require 
twice the number of yards from the roll; if only one-third 
of the width, then 3 times the number of yards. We may 
tabulate thus: 

Running yards. Width in inches. 

40 72 

80 36 

160 18 

320 9 

Clearly we cannot select a pair of terms from one column and 
equate them to the corresponding pair from the other. One 
pair has to be inverted, e.g. 



Thus when one quantity varies inversely as another, the in- 



RATIO AND PROPORTION 105 

version of one ratio (it matters not which) is necessary before 
equating. 

Teach the boys to distinguish between direct and inverse 
proportion by asking themselves whether when one quantity 
increases the other increases or decreases, and to distinguish 
them on paper by pointing arrows in the same direction to 
indicate direct proportion, in opposite directions to indicate 
inverse proportion. 

3 Ib. of tea cost 8s.; how many Ib. will cost 24$.? 

Lb. Shillings. 

1 3 8 i 

* x 24 * 

6 men can do a piece of work in 20 days\ how many could 
do it in 15 days? 

Men. Days. 

1 6 t 2 

* x I 15 

Boyle's Law is the best known example of inverse proportion 
in science, but in practical life examples of inverse proportion 
are much less common than those of direct, and the conse- 
quence is that very artificial examples are often invented to 
illustrate it. " Men and work " sums are often silly. " If 
it takes 20 men to build a house in 20 days ", more than 
one maker of an arithmetic book will ask us to believe that, 
as a logical consequence, 1000 men could build the house in 
I of one day. 

Never use the old form of proportional statement, : :: : . 
A common (and meaningless) form of statement sometimes 
found in a boy's exercise book is 

36 : 40 :: 24. 

Always let the equated ratios consist of two fractions, and 
make the boy realize that the particular position of the x 
(first, second, third, or fourth place) is entirely without 
significance. 



106 CRAFTSMANSHIP IN MATHEMATICS 



Examples Acceptable and Unacceptable 

We give two more examples. 

1. A clock which was 1$ minutes fast at 10.45 p.m. on 
2nd December was 8 minutes slow at 9 a.m. on 1th December. 
When was it exactly right? 

This problem, like most other problems, requires a 
preliminary discussion. By judicious questioning, help the 
boys to cast it in a simpler form: 

A slow-going clock loses 9^ of its false minutes in 106J true 
hours. In how many hours will it lose 1$ of its false minutes? 

" Minutes " lost by slow clock during Hours of true clock 

; ; 106i i 

i-J- x 

9J ^ lOGJ 

ij *T* 

85 _ 425 
" 13 " 4*"* 

JL = _^ 

" 13 ~ 4i" 
/. x = 161, 

i.e. the slow clock was right 16J hours after 10.45 p.m. on 
2nd December, i.e. at 3 p.m. on 3rd December. 

2. It takes 8 men 6 days to mow a field of grass. How 
long would it take 20 men to do it? 

Days Men 

f 




8 
20 



6 = 20 

x ~~ 8" 

x = 2f . 



But although 2| days is the orthodox answer, the time would 
really be rather less. Men mowing a field for 6 days would 



RATIO AND PROPORTION 107 

find, in the growing season, the work much harder on for 
example, the sixth day than on the first, so that the amount 
of grass cut would not be equally distributed over the 6 days. 
The answer as calculated is but a rough approximation. 
Writers of textbooks, and some examiners, are so often out 
of touch with practical life that it may be useful to append 
a few absurd questions of the type supposed to be examples 
of Ratio and Proportion: 

1. It takes 3 minutes to boil 5 eggs. How long would it 
take to boil 6 eggs? 

2. A man rides a bicycle at the rate of 20 miles an hour. 
How far could he travel in 92| hours? 

3. My salary is 500 a year and I save 50 a year. How 
long shall I take to save 10,000? 

4. My brother weighed 24 Ib. when he was 3 years old. 
How much will he weigh when he is 45 years old? 

5. A rope stretches | in. when loaded with 1 cwt. How 
much will it stretch when loaded with 10 tons? 

6. It cost 1 to dig and line a well 2 ft. deep. How 
much will it cost to dig and line a well 100 ft. deep? 

7. A stone dropped down an empty well 16 ft. deep 
reaches the bottom in 1 second. What is the depth of another 
well, if a stone takes 5 seconds to reach the bottom? 

Another point: if the answer to problems concerning men 
and work comes out to, say 4J men; instruct the boy to say 
5 men, with an explanatory note. 

Until a boy is thoroughly well grounded in Ratio and 
Proportion, the formal statement of the ratios is desirable. 
But at least the abler boys in the top Forms may be allowed 
to do as mathematicians themselves do write down the 
odd term and multiply at once by the fraction determined 
by the ratio of the other two terms. 

3 1 Ib. of tea cost llf shillings; find the cost of 1\ Ib. 

Cost = llfs. x -f . 
3 s 



108 CRAFTSMANSHIP IN MATHEMATICS 

" Compound " Proportion 

" Double" or " Compound " "Rule of Three ". 

For the most part the typical sums given by the textbooks 
to illustrate this " rule " have little relation to practical life. 
Occasionally they are legitimate enough, and then they may 
be regarded as just a simple extension of the simpler two- 
ratio examples already considered. The terms may be 
arranged in their natural pairs, converted into ratios, these 
marked (with arrows) direct or inverse, then multiplied out. 
If 16 cwt. are carried 63 miles for 6, 6$., what weight 
can be carried 112 miles for 2, 16s.? 

cost in 
cwt. miles. shillings 

\ x A 112 I 56 

* 16 I 63 * 126 



x 

112 126 



= 4. 



Here is another, one of the commoner types, taken from one 
of the best of the textbooks: If 36 men working 8 hours a 
day for 16 days can dig a trench 72 yd. long, 18 ft. wide, 12/J. 
deep, in how many days can 32 men working 12 hours a day 
dig a trench 64 yd. long, 27 ft. wide, and 18 ft. deep? 

The example is not practicable. Men working 12 hours 
a day can not do 1-J times as much work as men working 
8 hours a day. The cost of digging a trench 18 ft. deep is 
more than \\ times the cost of digging one 12 ft. deep. The 
deeper the trench the more expensive it is to get out the 
excavated earth. The cost does not necessarily vary as the 
width of the trench; if timbering the sidfcs is necessary (a 
serious additional item of expenditure), a little extra width 
would not add appreciably to the cost. But more than this: 
for excavation work, steam navvies have largely replaced 
manual labour. 

So it is with a large number of the textbook exercises: 



RATIO AND PROPORTION 109 

they have no relation to practical life. Here is one more, 
from a really excellent textbook. If 10 cannon which fire 
3 rounds in 5 minutes kill 270 men in 1| hours, how many cannon 
which fire 5 rounds in 6 minutes will kill 500 men in 1 hour? 
Did the man who made up this problem claim to be a mathe- 
matician, or a soldier, or a humorist? It is a shocking thing 
that school boys are made to waste their time over the pretence 
of " solving " problems of this kind. 



CHAPTER XIII 

Commercial Arithmetic 

No branch of arithmetic is more important, and yet it 
need not take up a very great deal of time. For the most 
part, the work consists of the application of principles, already 
learnt, to business relations in practical life. Once the boy 
grasps the inner nature of the business relation, the arith- 
metic should give him little trouble. But " hard " sums, 
especially sums involving great masses of figures, are rarely 
if ever necessary. Give ample practice in working easy 
exercises and so make the boy thoroughly familiar with the 
A B C of commercial life. 

Percentages 

Teach the meaning of " per cent " thoroughly. We 
require a numerical standard of reference of some kind, and 
the number 100 has been accepted as the most convenient, 
though any other would do instead. It is a disadvantage 
that 100 is not divisible by 3. 

5 per cent means 5 per 100 or 1 jfo. Drive this cardinal 
fact well home: everything hangs upon it. 5 per cent of 



110 CRAFTSMANSHIP IN MATHEMATICS 



1 = y^ of 20*. = 1*.; 2J per cent of 1 = 6df. Let these 
two results be the pegs of plenty of mental arithmetic; e.g. 
1\ per cent of 1 = 3 times 6rf. = Is. 6d., and 7 per cent 
of 20 = 1$. 6d. X 20 = 30^. ; and so on. 

Representative percentage numbers (as they are usefully 
called) is the next thing to drive home. When we buy a 
thing, it may be assumed that we buy it at the standard 
price which is represented by 100. If we sell the thing at 
10 per cent profit, we sell it at a price represented by 110; 
if we sell it at a loss of 15 per cent, we sell it at a price 
represented by the number 85. This notion is of fundamental 
importance. The majority of exercises grouped under the 
term " percentages " or " profit and loss " are cases of 
simple proportion, the two terms of one ratio consisting of 
money and the two terms of the other ratio consisting of 
representative percentage numbers. 

How much is 12f per cent of 566, 13*. 4rf.? 



Direct proportion example (common): If a debt after a 
deduction of 3 per cent becomes 210, 3s. 4rf., what would it 
have become if a deduction of 4 per cent had been made? 

Reduced debts. 

97 , 210* 

96 * x 

13 16 



i i 

= 208. 

Inverse proportion example (comparatively rare): A 
fruiterer buys shilling baskets of cherries, 30 in a basket. He 
also sells them at a shilling a basket^ but 24 in a basket. What 
profit per cent does he make? 



COMMERCIAL ARITHMETIC 111 

The smaller the number of cherries he sells in a basket, 
the larger his profit. Hence the proportion is inverse. 



Cherries per 
Basket. 


Representative 
% Nos. 


1 30 
* 24 


A 100 

' X 


x = 


" 



= 125. 

This representative percentage number shows that the profit 
is 25 per cent. (Strictly, the answer is not right, as no allow- 
ance is made for the necessary purchase of new baskets.) 

Another inverse proportion example: If eggs are bought 
at 21 for Is., how many must be sold for a guinea, to give a 
profit of 12| per cent? 

The selling price is represented by 112|, a number 
greater than 100; the number of eggs sold for a guinea must 
be smaller than the number bought for a guinea. Hence 
the proportion is inverse. 

Ar r 01 / Representative 

No. for 21/- 



441 I 100 

x * 112J 

.- 441 X^ 
112J 

=441 X f 
= 392. 



Simple Interest 



The kind of examples really necessary should cause 
little trouble. Even a slow boy readily understands the main 
principles. As soon as he has learnt what 5 per cent per 
annum means, he can follow this reasoning: 



112 CRAFTSMANSHIP IN MATHEMATICS 

Interest on 100 at 5 per cent per annum for 1 year 



Interest on seven times 100, i.e. on 700 for 1 year 

= 700 X Ttld- 
Interest on 700 for 3 years = 700 X T |o X 3. 

There is now an excellent opportunity for establishing a 
simple algebraic formula: 

Let I = Interest. 
P = Principal. 
R = Rate per cent. 
,, T = Time in years. 

Then I- PxT J*R^ PTR 
men i 1Q() - 10Q . 

The technical term " amount " should also be explained: 
A-P + I. 

As interest is usually paid half-yearly, " 5 per cent per 
annum " (as in the case of Government Stock) generally 
represents rather more than its nominal value. This should 
be explained. 

The use of the formula is quite legitimate, provided the 
boy has learnt to establish it from first principles; and equally 
he may be allowed to deduce the subsidiary formulae, arguing 
in this way: 

Since from first principles 
i _ PTR 

loo' 

/. I X 100 = PTR. 

P = L2L1? . T - L* -I??- R = * x 1QO 

" ~ TR ' PR ; PT " 

But in practical life these subsidiary problems (to find P or 
T or R) are very rarely wanted, and it is not worth while 
to let boys waste time over working a large number. An 
occasional example, mainly to give facility in the use of the 
formula, is enough. 



COMMERCIAL ARITHMETIC 113 



Compound Interest 

It is enough to tell a boy to find out what will be due to 
him if he places in the Bank 100 on deposit and allows 
it to remain there for 2 or 3 years, the interest, say at " 4 
per cent ", being undrawn. Two minutes of explanation 
will show him how to work the sum, each half-year's interest 
being added to the Principal as it becomes due. A little 
later on, instruction will be necessary as to shorter procedure 
in calculation, but to give up time to the working of numerous 
examples is inadvisable. Bankers never work compound 
interest sums: they merely refer to ready-made tables, 
prepared by mathematical hacks for all the world to use. 
Do not let the boys waste time over such useless work, 
especially as the time is so badly wanted for other things. 
On the other hand, see that they really do understand main 
principles, and can readily apply them to simple cases. 

The subject may, of course, be resumed in the Fifth or 
Sixth Form, should the general mathematical theory of 
interest and annuities be taken up. 



Present Worth and Discount 

Here again the principles are important and are very 
easily mastered. Their use may be amply illustrated by 
reference to a few easy examples. Do not forget to give a 
clear explanation of Bills of Exchange and Promissory 
Notes. 

The boy already knows from his interest sums that 

Amount =? Principal + Interest. 

In Discount sums, three new terms are used, and really they 
are identical with the three just mentioned: 

Sum Due = Present Worth + Discount. 

(1291) 9 



114 



CRAFTSMANSHIP IN MATHEMATICS 



Here are two exactly analogous examples in direct proportion 
of the normal type. 

What is the Principal that 
will produce an Amount of 840 
in 3 years at 4 per cent? 

When we have found the 
Principal, we can subtract it from 
the Amount, and so obtain the 
Interest. 



Principal, 
x 



Amount. 
840 



What is the Present Value of 
840, the Sum Due at the end 
of 3 years, the interest being 4 
per cent? 

When we have found the 
Present Value, we can subtract 
it from the Sum Due, and so 
obtain the Discount. 



Present Value. 

x 



Sum Due. 
840 



But there is only one term for each of the Ratios. Where are 
the others? We have to invent them. 

We do not know the value of x, the Principal that will 
amount to 840 in 3 years at 4 per cent. 

We do not know the value of x, the Present Value of the 
sum 840 due in 3 years at 4 per cent. 

But we may take any sum we please and invest it for 3 years 
at 4 per cent. 100 is as good a sum as any. 

100 invested for 3 years at 4 per cent yields 12 interest. 

Thus 100 is the Principal that Amounts to 112 in 
3 years at 4 per cent, and 100 is the Present Value of 112, 
the sum due in 3 years at 4 per cent. Now we may complete 
our Ratios. 



Principals. Amounts. 
* 100 ^ 112 



x 

ioo 



840 
If2' 



15 



= 750 (Principal) . 

Interest (if required) 
- 840 - 750 = 90. 



Sums Due. 
I 840 
* 112 



Present Values. 

4x 
100 

x __ 840 
100 112* 

15 

/. * = 100 x |g 

2 
= 750 (Present Value). 

Discount (if required) 
= 840 - 750 = 90. 



COMMERCIAL ARITHMETIC 115 

Remind the boy that this True discount is never heard of 
in practice. The Bill Broker's discount, which he deducts, 
is really the interest on the whole Sum Due. It is exactly the 
same as calculating Interest on the Amount instead of on the 
Principal, a thing the banker would (naturally) never dream 
of doing. Let the boy compare the two things, and see for 
himself that when the Banker deducts interest on the Sum 
Due instead of on the Present Value, the customer receives, 
as Present Value, a sum rather less than by arithmetic he 
is entitled to. 

Exercises in Present Value and Discount are hardly 
worth doing, unless they are very simple and can be done 
quickly. 

Stocks and Shares 

Nothing is more important in arithmetic than a working 
knowledge of stocks and shares and of financial operations. 
Whatever views political extremists may take of a roseate 
financial future, we have to deal with the hard facts of the 
present day, when it behoves every member of society to 
save, and to invest his savings. 

But do not make boys waste their time by working through 
the useless examples on stocks and shares given in many of 
the older textbooks. 

The first stile for the boy to get over is the distinction 
between stock and money, and there is no better way than 
to turn the whole class into an imaginary Limited Liability 
Company with its own Directors. To play a game of this 
kind is worth while. Let the Directors draw up a simple 
Prospectus and invite subscriptions at par. A week later 
let the Directors report some disaster perhaps the destruc- 
tion of property by fire and an inevitable fall in the expected 
interest. Some shareholders will become anxious and will 
be willing to sell at 90 or even lower. And so on. A little 
reality of this kind is worth ten times the value of a long 
sermon on the subject. If a boy pays 100 (any sort of paper 



116 CRAFTSMANSHIP IN MATHEMATICS 

token will do) and receives a Certificate for 100 1 shares, 
and then has to part with his shares at, say, 18$. each, a 
sense of reality is brought home to him. Perplexity about 
stocks and shares is almost always due to a hazy understanding 
of the reality which underlies it all. As always, the trouble 
is with the slower boys. The quicker boys pick up the threads 
readily enough. 

There are numerous facts for the boys to understand and 
remember, as well as sums to work. Explain the nature of 
debentures, preference shares, the different kinds of ordinary 
shares, their relative value and relative safety. Warn the 
boys never to invest without taking advice, and never in any 
circumstances to invest in a new flotation. Explain " gilt- 
edged " securities, and point out the relative safety of Govern- 
ment stock, though even this may fall seriously in value 
(compare the present price of Consols with the price fifty 
years ago). Insist that a large interest connotes a big risk, 
that financial greed spells disaster. Impress upon the boys 
that the financial world is full of sharks. 

The old days of a brokerage of ^ per cent have passed 
away, and thus many of the sums in the older text- 
books are out of date. Stockbrokers' charges now include 
Government Stamp Duty, Company's Registration fee, 
and Contract Stamp. Give the class a short table of 
charges to be entered in their notebooks, for permanent 
reference, e.g. 

Purchases 50+ to 75, total charges 18s. 3d. 
Purchases 75+ to 100, total charges 1, 3s. 3rf., 
and so on. 

All ordinary " examples " in stocks and shares are in- 
stances of simple proportion (nearly always direct: there is 
little point in puzzling young boys with the rule " the amount 
of stock held is inversely proportional to the price "), and 
they call for no comment. 

Examples on the purchase and on the sale of small amounts 
of stock and small numbers of shares are the only exercises 



COMMERCIAL ARITHMETIC 117 

that need be given. Let the exercises be typical of those 
that in practical life the average man engages in. 



Other Commercial Work 

Rates and Taxes. A simple explanation of and a variety 
of exercises in these are of great importance. Explain the 
increase in both rates and taxes during the last few years. 
Distinguish carefully between expenditure by the Govern- 
ment and expenditure by Local Authorities, and show why 
both kinds of expenditure are inevitable. Explain how taxes 
are imposed and how rates are levied. Let exercises be easy, 
but devise them to illustrate principles and to give an inner 
meaning to things. " Rateable values " is another thing to 
be explained. 

And there are numerous other things, of which it behoves 
every intelligent person to have at least an elementary know- 
ledge, things which only a mathematical teacher can handle 
effectively. We mention a few: Income Tax and its assess- 
ment, its schedules, its forms and the correct method of 
filling them up; rent, house purchase, mortgages; the raising 
of loans by public bodies and by private persons; insurance 
of all kinds, especially life insurance, Health and Unemploy- 
ment insurance; policies (especially " all-in " policies) and 
premiums; pensions, annuities, the keeping of personal 
accounts, thrift, household economics; banks and saving; 
the Post Office bank and National Saving Certificates; co- 
operative stores and their financial basis; building societies; 
insurance tables and how to read them (a Sixth Form ought 
to have some knowledge of their actuarial basis). There 
are numerous tables of very useful kinds in Whitaker that 
every boy ought to be made to understand, and by means of 
them an arithmetic teacher may devise exercises of a very 
valuable kind. 

A particularly useful syllabus on the arithmetic of 
citizenship is given in the appendix of the 1928 Report of 



118 CRAFTSMANSHIP IN MATHEMATICS 

the Girls' Schools Sub-committee of the Mathematical 
Association. 

Books on arithmetic to consult: 

1. The Psychology of Arithmetic, Thorndike. 

2. Lecons d'Arithme'tique, Tannery (Armand Colin). 

3. The Teaching of the Essentials of Arithmetic, Ballard. 

4. The Tutorial Arithmetic, Workman. 

5. The Groundwork of Arithmetic, Punnett. 

6. The Small Investor, Parkinson. 



CHAPTER XIV 

Mensuration 

Simple Formulae 

Easy problems involving actual measurements will be 
embodied in the mathematical course for children below the 
age of 11, by which time^ boy^ught to be familiar with the 
mensuration of rectangular areas and rectangular solids and 
to be able to work easy conversion (reduction) sums in linear, 
square, and cubic measures. He ought also to have learnt 
to measure up the area of the classroom floor and walls, and 
to express his results in formula fashion) e.g. area of floor 
= / X b\ area of the four walls = 2(1 + b)h 

\He should now be taught, if he has not been taught 
before, to make paper models of cubes and cuboids, and 
from a consideration of the " developed " surfaces of these, 
laid out in the form of " nets ", to devise formulae for cal- 
culating the areas;) e.g. of a cube, 6/ 2 ; of a square prism, 
4/z/ + 2a 2 or 2a(2l + a)\ of a brick, 2(lb + It + bt). The 
memorizing of these formulae is not worth while, but they 
are worth working out as generalizations from particular 
examples; and when, once more, numerical values are 
assigned to them, it makes early algebra very real.j) 



MENSURATION 119 

The Papering of Rooms 

Some attention must be given to the stock problems 
on the papering of rooms, but it is not worth while to take 
time over measuring up doors, windows, and fire-places; 
assume that the walls are unbroken planes, and the room 
rectangular. Nor is it worth while to divide the perimeter 
of the room by 21 in. to find the necessary number of strips 
of paper. There is bound to be a good deal of paper wasted, 
especially if the pattern is elaborate. Hence it is enough to 
take the total wall area 2(1 ~\~ b)h, divide this by the area of 
one roll of paper, 36 ft. X If ft. or 63 sq. ft. or 9 sq. yd., 
and so obtain the necessary number of rolls. If the answer 
comes out to 13 rolls, evidently 14 are wanted, perhaps 
15 because of waste; perhaps 13 or even 12 would do, because 
of windows, doors, &c. A paper-hanger never measures up 
a room with any degree of accuracy; his estimate is very 
rough and always done by rule of thumb. There is really 
no point in giving boys such problems to work, especially 
when it is remembered what a large number of problems, 
depending on accurate measurements, may be culled from 
the boys' physics course. 

The Carpeting of Floors 

The carpeting of floors is generally considered to give 
an easier type of problem than the papering of walls, but 
the problem in practice is a little tricky. If from an ordinary 
27-in. wide roll a carpet has to be made up to fit the usual 
rectangular room, it is unlikely that the width of the room 
is an exact multiple of 27 in., in which case the last of the 
strips cut off the roll will be too wide, and there will be 
waste; and yet the whole of that strip will have to be pur- 
chased, as the pattern cannot be " matched ". If the carpet 
is plain, and the purchaser does not object to patching, 
then the exact amount required may be cut from the roll, 



120 CRAFTSMANSHIP IN MATHEMATICS 

though the vendor might not agree to cut to the small fraction 
of a yard. 

Consider a floor 18' X 12', and carpet 2' 3" wide. 

1. Let the carpet be plain (patternless). Area of floor 
= 24 sq. yd. Required number of running yards from the 
roll 24 -r f = 32. This will give 5 strips, each 6 yd. 
long, and 2 running yards (a piece 6' X 2' 3") over. This 
strip of 6' X 2' 3" will have to be cut up to cover a space 
18' by I)", so that it will be cut into 3 pieces each 6' long, 
placed end to end, the width of these being 9". 

2. Let the carpet show a design, the width being the same 
as before. Evidently at least 6 strips, each 6 yd. long, or 
36 yd. in all, must be cut from the roll. It is highly im- 
probable that, if the strips are cut to exact length, they would 
match when laid side by side. There would be a good deal 
of waste, depending on the size of the design. The problem 
cannot be brought within the scope of classroom arithmetic: 
all the factors are not available. 

3. A more practical problem for the classroom is to 
estimate the amount of plain carpet required to cover a room 
of given size with a minimum number of complete strips, 
allowing the surplus width to determine an equal all-round 
border (to be stained or covered with linoleum). For instance, 
the 5 strips above mentioned would leave a surplus width of 9". 
If the strips are placed together centrally, there will be a width 
of 4|" to spare at each end of the room. Hence we must 
arrange for a complete border of 4|-" all round the room. Thus 
the 5 strips will not now be 18' long, but 17' 3" long, and the 
amount to be cut from the roll will be 17' 3" by 5 (28| yd.). 

Thus the area of the room = 18' X 12'; of the carpet, 
17' 3" X 11' 3"; and the required number of running yards 
from the 2' 3" wide roll - 28|. 

, If the whole floor had been covered, and patching was 
allowable, the number of running yards required = 32; if 
patching was not allowable, the number of running yards = 36 
(leaving a waste piece 6 yd. long and 18" wide, with an area 
of 3 sq. yd., equal to 4 running yards). 



MENSURATION 121 

Does it not all come round to this that these mensuration 
problems concerning wall-paper and carpet are rather futile, 
especially when whole chapters in arithmetic books are 
devoted to them? Children are much better employed in 
mensuration problems that really do enter into the practical 
business of life. 

Border Areas 

Make these a matter of subtraction, whenever possible 
as, for instance, in estimating the area of a garden path 
4' wide between a rectangular lawn and the rectangular 
garden wall the garden being 108' X 72'. 

Area = {(108 X 72) - (100 X 64)} sq. ft. 
Do not allow boys to find the area of the path piecemeal. 

Rectangular Solids 

For the mensuration of these, boys can, with very little 
help, establish the necessary simple formulae and interpret 
them in some brief form of words easily remembered. Prob- 
lems on the excavation of trenches, the cubical content of 
cisterns, the air space of school dormitories, and the like, 
will readily occur to the teacher. The cubical content of a 
solid " shell " (e.g. of iron in a cistern, of stone in a rectangular 
trough) should, whenever possible, be made a problem of 
subtraction. Example: Find the weight of a stone trough 
6" thick, external dimensions 10' X 3' X 2' 6", the weight of 
stone being \\ cwt. to 1 cubic foot. 

No. of c. ft. of stone = (10 X 3 X 2J) - (9 X 2 X 2) 

= 75-36 

= 39. 
Weight = 1J cwt. X 39 = 2 tons, 18| cwt. 

A boy should never attempt to cube up the stone piecemeal. 

If a gasholder (" gasometer ") at an ordinary gas-works is 

made the subject of a mensuration problem, remember that 



122 CRAFTSMANSHIP IN MATHEMATICS 

(1) a gasholder has no bottom, (2) its top is not flat. Not all 
writers of arithmetic books seem to realize this. 

Mensuration beyond the very elementary stage is best 
associated, primarily, with the geometry rather than with the 
arithmetic. 



CHAPTER XV 

The Beginnings of Algebra 

Informal Beginnings 

Regarded as simple generalized arithmetic, algebra will 
have been begun at the age of about 9 or 10. Quite young 
boys will have measured up rectangular areas and will have 
learnt to express intelligently the meaning of the formula 
A = / X b. In their lessons on physical measurements, 
rather older boys will probably have evaluated 77, 2?7R, 
?rR 2 ; in their arithmetic lessons they will have established 

PTR 

the formula I = ; in their first lessons on Ratio and 

x 1 

Proportion, they will have learnt the significance of - = 

6 14 

and will have obtained the first notions of an equation. 
Formally, algebra will not have been begun; informally, 
foundations will have been laid. 

Never begin the teaching of the subject according to the 
sequence of the older textbooks. The difficult examples in 
mechanical work so often given on the first four rules, on 
H.C.F.s and L.CJVI.s, on fractions, &c., are not only calculated 
to make boys hate the subject but are wholly unprofitable 
either at an early stage or later. 

Suppose you are asking questions in mental arithmetic 
to a class of boys 10 or 11 years of age, and you suddenly 
spring upon them the sum, " add together all the numbers 



THE BEGINNINGS OF ALGEBRA 123 

from 1 to 100." " We cannot do it, sir/'" Well, let us 
try. Let us first take an easier sum of the same kind: add 
together all the numbers from 1 to 12. We will do it in this 
way: 

" Add together the first number and the last, 1 and 

12? 13. 
" Add the 2nd from the beginning and the 2nd from 

the end, 2 and 11? 13. 
" Add the 3rd from the beginning and the 3rd from 

the end, 3 and 10? 13. 
" 4 and 9? 13; 5 and 8? 13; 6 and 7? 13. 

Now we have included them all. How many 13's? 6. What 
aresix!3's? 78. This 78 must be the answer to the question. " { 
Smiles of agreement. 

" Now let us make up a little formula that we can use 
for similar sums: How did we obtain the first 13? We added 
together the first number and the last. 

" What is the first letter of the word first? f. 
" What is the first letter of the word last? I. 
" How can we show the sum of/ and /? / + /. 
" How far along the line from 1 and 12 was our mul- 
tiplier, 6? Half-way. 
" What is the first letter of the word half? h. 

" Now I will show you how to write down/ + / multiplied 
by h" Then follows a brief explanation, and A(/+ 0- 

" Now let us work the harder sum, 1 to 100. 

"/= ? 1; "/= ? 100; "A = ?" 50. 

" .'. h(f + /) = 50(1 + 100) = (50 X 101) = 5050. 

Now add together all the numbers from 1 to 1000." And 
so on. " I have been giving you an algebra lesson which I 
sometimes give to boys 2 or 3 years older. An interesting 
subject, isn't it?" Yes. " And useful?" Yes. 

Or we might begin straight away with problems producing 
equations. First notions of an equation will already have 



124 CRAFTSMANSHIP IN MATHEMATICS 

been given in arithmetic. By means of a few easy exercises, 
revise the principle that the two sides of an equation may 
be added to, diminished, multiplied, or divided, by any 
number we please, provided that the two sides are treated 
exactly alike. The rule of cross multiplication should also 
be revised. But naturally at thi$ stage no equation should 
be given with a binomial in a denominator. 

Here are two examples, in a teacher's own phraseology 
(summarized, except that, to save space, his many admirably 
framed questions are omitted), once taken with a class of 
beginners of 11. 

The sum of 50 is to be divided among 2 men, 3 women, 
and 4 boys, so that each man shall have twice as much as each 
woman and each woman 3 times as much as each boy. Required 
the share of each. 

" In sums of this kind it is always well to consider first 
the person who is to have the least, in this case a boy. 

Let x be the number of 's in each boy's share. 
Then 3x is the number of 's in each woman's share. 
And Qx is the number of 's in each man's share. 
Hence we have, in 's, 

The share of the 4 boys = 4#. 
The share of the 3 women = 9x. 
The share of the 2 men = I2x. 

But the sum of all these shares amounts to 50. 

.'. 4* -f- 9* + 12* = 50, 
25* - 50, 



/ want to divide some nuts among a certain number of 
boys. If I give 4 nuts to each boy, I shall have 2 nuts to spare; 
if I give 3 to each boy, I shall have 8 to spare. How many boys 
are there? 

" Let x be the number of boys. 

" There are two parts to this problem, both beginning 
with the word if\ each part enables us to write down the 



THE BEGINNINGS OF ALGEBRA 125 

total number of nuts> though not exactly as in arithmetic, 
because we have to use x. 

" 1. If I give 4 nuts to each of x boys, I give away 4# 
nuts. But the total number of nuts is 2 more than that. 

.'. the total number of nuts = 4# -f- 2. 

" 2. If I give 3 nuts to each of x boys, I give away 3x 
nuts. But the total number of nuts is 8 more than that. 

/. the total number of tints -- 3# -f 8. 

.'. 4x + 2 = 3jc + 8, 

.'. * + 2 = 8, 



This class of boys subsequently spent the next three or four 
lessons in working through a chapter of problems (some of them 
pretty difficult) producing equations. They gained confidence 
quickly, and from the first looked upon their new subject as 
interesting and useful. 

Formal Beginnings. Signs as "Direction Posts" 

Of course the time comes when algebra must be treated 
formally. There are certain fundamental difficulties that have 
to be faced, and, of these, algebraic subtraction is to beginners 
a difficulty of a serious kind. 

This does not mean that boys need be taken through the 
elaborate subtraction sums of the older textbooks. It means 
that they have to be taught the inner meaning of, say, " take 
3# from 2# ". To the boy, what does that meant At 
first it can mean nothing but juggling with arithmetical 
values, juggling of which he is naturally suspicious. 

Consider, first, what the boy has already done (or ought 
to have done) in his arithmetic. He is familiar with this kind 
of sum. 17 _ 6 __ 5 + 3 _ 9 + 14 

= 17 + 3 + 14-6-5-9 
= (17 + 3 + 14) - (6 + 5 + 9) 
= 34-20 
= 14. 



126 CRAFTSMANSHIP IN MATHEMATICS 

He has been taught to collect up his numbers in this way, 
and to realize that the plan of adding together all the minus 
numbers and taking them away "in a lump" is a much better 
plan than taking them away separately. Thus he sees that 

34-6-5-9 

must be the same as 

34 - (6 -f 5 + 9), 

and he therefore gets a first notion of the effect of a minus sign 
before a pair of brackets. Still, the work so far is wholly 
numerical, and nothing more. If the sum had been 

20- 34 

he would probably have been taught to prefix a + sign to 
the 20: 

+ 20 - 34, 

to take the difference between the 20 and the 34, and to prefix 
the sign of the larger number (34); thus, 14. Naturally 
the boy would call it a " subtraction " sum, and would say 
that the 14 is 'Mess than nothing "I The teacher might, 
however, allow the use of this illogical expression provisionally, 
comparing it with debts as against assets. 

But the boy must soon come to grips with the funda- 
mental algebraic notion of direction, as well as of numerical 
value. 

In arithmetic always, in algebra commonly, the + sig n 
before an initial term is omitted. But in the early stages of 
algebra it is advisable that it be consistently written. 

In algebra it is necessary to have some means of distin- 
guishing direction to the right from the opposite direction 
to the lefty and direction upwards from the opposite direction 
downwards. The opposed signs + and are used for this 
purpose. These signs are the algebraic signposts or direction 
posts; the two signs direct the numbers to which they are 
attached. It has been agreed that direction upwards and to 
the right shall be called a + (positive) direction, and direction 



THE BEGINNINGS OF ALGEBRA 



127 



downwards and to the left a (negative) direction. The 
converse would have done equally well, but the decision has 
been universally accepted. It is just a convention. If the 
boy asks why? tell him there is no answer. 

Consider the centigrade thermometer, with the freezing- 
point marked 0. If the temperature is 5 and rises 20, 
every boy knows that it rises to + 15; i.e. -5 + 20 = +15; 
also that if the temperature is, say, +10, and falls 25, 
it falls to 15, i.e. +10 25 15. The results may be 
obtained by actual counting, upwards or downwards, on the 
scale. Upward counting means adding + numbers; down- 
ward counting means adding numbers. The thermometer 
provides an excellent means of giving a first lesson on algebraic 
direction. 

Addition and Subtraction 

Now consider a more general case. We will choose a 
horizontal scale, with + numbers and numbers to the 
right and left, respectively, of a zero. 

Adding + quantities means counting to the right. 
Adding - quantities means counting to the left. 



__ . ___ | j p-j 

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 



I I 1 I I I I I I I 
+1 +2 +3 4-4 +5 +6 +7 +8 +9 +10 



Four addition sums: 





Addition Sums. 


Starting Point 
on Scale. 


Count or 
Add on Scale. 


Direction, 
Ror L. 


New Point on 
Scale and /. Ans. 


i 


(+5) + (+3) 


+3 


+5 


R 


+ 8 


ii 


( + 5) + (-3) 


-3 


+ 5 


R 


+2 


iii 


(-5) + (+3) 


+3 


-5 


L 


2 


iv 


(-5) + (-3) 


-3 


-5 


L 


-8 



Four subtraction sums. Where we have to work a 
subtraction sum, say 12 7, we may work it by asking what 



128 



CRAFTSMANSHIP IN MATHEMATICS 



we must add to 7 to make 12. Thus in the four subtraction 
sums below we may say, 

(i) What must we add to (+3) to make (+5)? 
(ii) ,, ,, (-3) (+5)? 
(Hi) ,, (+3) (-5)? 
(iv) (-3) (-5)? 





Subtraction Sums. 


Starting Point 
on Scale. 


Scale Point 
to count to. 


Direction, 
R or L. 


Number of Points 
counted 

= Answer. 


i 


( + 5) - ( i 3) 


+ 3 


-1-5 


R 


+ 2 


ii 


( + 5) - (-3) 


-3 


-1-5 


R 


+ 8 


iii 


(-5) - ( + 3) 


+ 3 


-5 


L 


-8 


iv 


(-5) - (-3) 


-3 


-5 


L 


-2 










i 



Let the boys now examine the two groups of answers and 
note from them that: 



( + 5) + ( + 3) - 

( + 5) + (-3) = 

(-5) + ( + 3) = 

(-6) + (-3) = 



( + 6) -(-3) 
( + 5) -(4-3) 
(-5) -(-3) 
(-5) -(4-3) 



They thus learn that in every case we can turn a subtraction 
sum into an addition sum merely by changing the sign of the 
subtrahend. 

They ought now to understand that in arithmetical addition 
the total is increased by each term added; that in algebraic 
addition the numbers indicate movements or distances back- 
wards or forwards along a line from a zero point or " origin ". 

They ought also to see that in algebraic addition we may 
drop the sign which separates the components, and deal 
with the components in accordance with their own signs, e.g. 

(4-5) + (4-3) -4-54-3 
(4-5) 4- (-3) = +5-3 
(-5) + ( + 3)- -5 + 3 
(_5) + (-3)= -5-3 



For algebraic subtraction, let them substitute algebraic 



THE BEGINNINGS OF ALGEBRA 129 

addition, at the same time always reversing the sign of the 
second term (subtrahend). Since the sum is now an addition 
sum, we may drop the connecting + sign as before: 

( + 5) - ( + 3) = ( + 5) + (~3) = +5 - 3 
( + 5) - (-3) = ( + 5) 4- ( + 3) -4-5-1-3 
(-5) - (4-3) - (-5) 4- (-3) = -5 - 3 
(-5) - (-3) - (-5) 4- (4-3) --64-3 

Examples: Add 11 x and 19*: 

4-17* - 19* = -2*. 
From 17* take 19*: 

4-17* - (-19*) - 17* 4- 19* = 4-36*. 

For the slow boys, indeed for all boys, the whole process 
crystallizes into three simple little rules: 

1. Addition sums. 

(i) Like signs: add, and prefix the same sign, 
(ii) Unlike signs: find the difference between the two 
numbers and prefix the sign of the larger. 

2. Subtraction sums. Reverse the sign of the second term 

(subtrahend) and treat the sum as an addition sum. 

Teachers are not always quite happy about this question of 
directed numbers, and often ask if it is not unwise even to 
make the attempt to deal with it, and if a statement of just the 
rules ought not to suffice. Of the answer I have no doubt. 
Boys who do not grasp the significance of directed numbers 
can never get to the bottom of their algebra; their work 
all through will inevitably be mechanical. Admittedly, 
however, the non-mathematical boy fails to understand, and 
for him the rules, as rules, must suffice. A Sets can and must 
master the difficulty, and I think B Sets too. But with C Sets, 
and especially with D Sets, be content to state the rules 
and to give the boys plenty of practice in them. Such boys 
will never make mathematicians, and nobody expects that they 
will. It is best to admit that the application of signs to com- 
ponent and resultant scale distances is a little too subtle 
for the non-mathematical boy. 

(E291) 10 



130 



CRAFTSMANSHIP IN MATHEMATICS 



M 



+200 




-5 -4 -3 -2 



+2 +3 



+5 



Multiplication 

Here again the rule of signs can be understood only by 
a clear grasp of the effect of direction. The usual train 
illustration is as good as any.* 

Graph the route of a train travelling northwards through O 
(say Oxford) at the rate of 40 miles an hour, and thus show 
the position of the train at all points on its journey. 

Let horizontal lengths to the right of MOM' measure 
times after train reaches O, and let the times be indicated 

by -f- numbers; and 
let those to the left 
of MOM' measure 
times before train 
reaches O, and let 
these times be indi- 
_ H cated by numbers. 
Let lengths above 
H'OH measure dis- 
tances north of O, 
and let these be indi- 
cated by + numbers; 
and let those below 
H'OH measure dis- 
tances south of O, 
and let these be indicated by numbers. 

We will mark the positions of the train 4 hours before 
reaching O and 4 hours after passing O. (The scales used 
are 5 mm. to 50 miles and 5 mm. to 1 hour.) At 40 miles 
an hour, the train must, at these times, be 160 miles short of 
O and 160 beyond O, respectively. Plot points P 2 and P l 
to show this. P 2 must be directly below 4 on H'H, and to 
the left of 160 on M'M; P l must be directly above +4 
on H'H and to the right of +160 on MM'. The line P 2 P a 
evidently passes through O, and represents the train route. 



P 2 (-I60) 



-60 



-ISO 



-200 

M' 

Fig. 14 



See Nunn, Teaching of Algebra, Chap. XVIII. 



THE BEGINNINGS OF ALGEBRA 



131 



Now how can we determine the two positions by cal- 
culation? 

We may utilize the formula d = vt (" distance = speed 
X time "), and by making the three symbols stand for 
directed numbers, the formula will give us information about 
the direction as well as the magnitude. Hence we must use 
the term velocity. Let velocity northwards (40 miles an hour) 
be considered -f. 

1. Position of train at P 2 : 

d = vt = ( + 40) x (-4) 

= -160 (as graphed) = 160 miles S. 

2. Position of train at P x : 

d = vt = ( + 40) x ( + 4) 

= +160 (as graphed) = 160 miles N. 

Now consider the train travelling southwards. Let velocity 
southwards (40 miles an hour) be considered negative ( -). 




3. Position of train at P 3 : 

d = vt = (-40) X (-4) 

= +160 (as graphed) =160 miles N. 

4. Position of train at P 4 : 

d = vt = (- 40) x ( + 4) 

= -160 (as graphed) = 160 miles S. 



132 CRAFTSMANSHIP IN MATHEMATICS 

Comparing the 4 results we have: 

( + 40) x ( + 4) = +160 

; + 40) x (-4) = -160 

(-40) X (-4) == +160 

(-40) X ( + 4) = -160 

This is enough, at this stage, to justify the sign rule for mul- 
tiplication. A more rigorous generalization may, if necessary, 
come later. 

(The boys should be made to see that the sloping lines in 
the above graphs do not graphically show the actual railway 
track, which is supposed to run due north-south.) 

It may be urged that the whole thing seems to be a 
little artificial. So it is. But the rule of signs is a universally 
accepted convention. The convention is perfectly self- 
consistent, and is easily justified, but by its nature it admits 
of no " proof ". 

Book to consult: The Teaching of Algebra, Nunn. 



CHAPTER XVI 

Algebra: Early Links with Arithmetic 
and Geometry 

Algebra and Arithmetic in Parallel 

Get the boys to see that an algebraic fraction is only a 
shorthand description of actual arithmetical fractions, and 
that there is really no difference in the treatment. The working 
processes are practically identical. 

The arithmetical fraction -^ may be written ?, 
which shows clearly that the denominator is greater by 3 
than the numerator. So does the fraction ^, and that 
is all it means. Thus in the fraction ~j, a + 7 represents 



ALGEBRA: EARLY LINKS 



133 



a single number; as in arithmetic, it must be moved as a 
whole from one place in the expression to another. In algebra, 
beginners sometimes forget this, and treat the parts of a 
binomial denominator separately. So with 2 or more bi- 
nomial denominators: for instance in - r>4"jirfi> x ^ 
and x + 11 express single numbers. 

Show a few corresponding arithmetical algebraic processes 
side by side. It helps the slower boys much. 

1. Let a = 4, 6 -= 7. 



28 

L 1 

28" 



28 



- 

a b 

b .,. - 

ttb ab 



2. Let a = 3, b = 4, c = 5 

1 + - 4 + - 5 
20 15 12 

= 3 4 2 6* 

~~ 60 60 60 
32 4. 42 + 52 



3. Let a = 7, b 4. 
_1 J^ i< 

7~- 4 7~+~i ' 49~~ 
= 1 1 _ 14 
~ 3 11 33 

- I 1 4- 3 _ U 
33 33 33 

- H + 3 ~ 14 

"33" 

= JO 

"" 33 

- 0. 



be 



4- 4- ^ 
ca a6 

6^ ^ 
^c a^ 



1 , 1 



2a 



a b a -\~ b a 2 b 2 



4- __1_ _ ___ 
a 2 - b* a* - b* 

-^a b 2a 



0. 



134 



CRAFTSMANSHIP IN MATHEMATICS 



There is little or no need to take fractions beyond quite simple 
binomial denominators. Denominators of a higher order are 
seldom required in practice. Hence all H.C.F.s and L.C.M.s 
should be evaluated by factorizing, exactly as in arithmetic. 
The principle of the cumbrous divisional processes for finding 
factors should be familiar to boys in A Sets, who, however, 
may be referred to their textbooks. Do not waste time over 
such things in class. 

Geometrical Illustrations 

Factors, multiplication, division, simple expansions, &c., 
should all, in the early stages, be illustrated geometrically, 
and thus be given a reality. When this reality is appreciated, 
but not before, the illustrations may be given up. Second 
power expressions should be consistently associated with areas. 
We append a few illustrative examples. 



2 


20 


20 


X 




4 


\ 20 * 4 


4* 



2 
2 ^ 


at 


> ab 


b* 



Fig. 1 6 

1. Compare the square of 24 (i.e. 20 + 4) with the square 
of a + b. 

24 a (a + 6) a 

= (20 + 4) 8 =fl 2 + 2a& + 6 2 . 

= 20 2 -f 2.20.4 + 4 2 . 

2. Compare 24x27, i.e. (20 + 4) (20 + 7), with 



ALGEBRA: EARLY LINKS 



135 



(20 + 4) (20 + 7) 

= 20 a + 20.7 + 20.4 + 4.7 
= 20 2 + 20(7 + 4) + 4.7. 



(a + 4) (a + 7) 
= a 2 + 7a + 4a + 4.7 
= a 2 + a(7 + 4) + 4.7. 



-20- 






20 2 


20 

X 




4 


20*7 


4 

X 

7 





2 


CL 


I 


a 


X 






4 


r 

\ 


a x 7 


4 
x 
7 



Fig. 17 



3. Show graphically that 

(2a + 56) (a + 36) = 2a 2 + Uab + 156 2 . 

o. a. b b b b b 



a 

b 
b 
b 



Fig. 1 8 

The result is seen at a glance. 



2 
a. 


2 

a 


ab 


ab 


. 


ab 


. 


I 

an 


cxb 


ab 


b 2 


b 2 


b 2 


b 2 


b 2 




db 


ob 


b ? 


b 2 


b 2 


b 2 


b 2 




ab 


ab 


b 2 


b 2 


b 2 


b 2 


b 2 


1 




?<, + 5^ 































136 CRAFTSMANSHIP IN MATHEMATICS 

4. Show graphically that (a 2) (a 3) = a 2 6a + 6. 



A 


* a -2 * 


/ 




D 


t 
a. S 

i 


F 






J 
a 


G 


E 








/ 










/ 








f 


B 


'* Q- 






C 













AE = (a - 2) (0 - 3), 
AC - a 2 , 
GC - 3a, 



. 19 

rcc\- l 
== ^<z, 

GC + FC - EC = 6fl - 6, 
AE - AC - (GF -f FC - EC) 



i.e. (a - 2) (a - 3) = a 2 - 5a -f 6. 
5. Show graphically that (a + 2) (a 3) a 2 a 6. 



4 


E / / 


3 


p 

R 


(a + 2) (a - 3> 






t 

a-3 


P 


K C 

F 


















q 


/ / / S 





Fig. 20 

(a + 2) (a - 3) = AC I = AK + PF 

= AK f EC I = AF - DQ - RS; 

i.e. (a + 2) (a 3) == a 2 a 6. 



ALGEBRA: EARLY LINKS 

6. Show graphically that 

(a + b + cf = a* + b* + c 2 + 2ab + 2ac + 2bc. 



137 





- a. *|_b-*|4C*. 


r 








a 


2 
CL 


ab 


ac 


5 


ab 


tf 


be 


[ 








I 

f 


ac 


be 


c 2 



Fig. 21 

The result is seen at a glance. 



CHAPTER XVII 

Graphs 

From the Column to the Locus 

Begin with column graphs, that is with mere verticals 
with the tops unconnected; say, the amount of gas consumed 
each week for a quarter, or the height of a barometer each 
morning for a week. Now join the tops of the columns, 
first by straight lines, then by a curved line. Do the straight 
lines teach anything? Which is likely to be the more correct, 
the straight lines or the curved line? Can intermediate columns 
be inserted, and, if so, what would they signify? Are there 
any cases where intermediate columns would be absurd? 

Now discuss the locus graph, as distinct from the column 



138 



CRAFTSMANSHIP IN MATHEMATICS 



graph. The barograph is a useful example especially for the 
contrast of a gentle slope and a steep slope, and hence as an 
introduction to a gradient and what it signifies. What does 
a chart of closely packed isobars signify? of open isobars? 
Or graph the vertical section of a piece of hilly country, by 
taking heights from an ordnance survey map. Here the gentle 
slope and the steep slope appeal at once, the closely packed 
isobars and the closely packed contours being akin. 




Dates (20 
of 1^ 
lnnings[ MAY 



The significance of a gentle gradient and of a steep 
gradient is fundamental. It is really the key to all subse- 
quent work. Let the boys graph their cricket scores for 
the previous summer term, and discuss the resulting 
gradients. Familiar and personal data of this kind often 
provoke animated discussion of a useful character. In the 
first lesson or two, much of the work can be done on the 
blackboard, exact numerical values playing only a minor 
part. Give the beginners a general notion of the graph and 
its significance. A few instances may be culled from chemistry 
and physics, say solution curves (common salt, with its 



GRAPHS 



139 



very slowly rising straight line; nitre, with its steep curve); 
the experimental results, with their subsequent pictorial 
illustrations, are always impressive. Other useful graphs 
from practical work are a straight-line graph from a loaded 
spiral spring, or from a F.-C. scale comparison; an inverse 
proportion graph, say a time-speed curve during a journey. 
Thus prepare the way for formal work. 



15 



penc 







5 

Ib 

Fig. 23 



A parcel post graph is easy of interpretation and, by 
its gradient of equal steps, leads on naturally from a column 
graph to a direct proportion graph. It may be called a 
" stepped " graph. There is a minimum charge of 6d. for 
any weight of parcel up to 2 Ib.; the charge is 9d. for any 
weight over 2 Ib. and up to 5 Ib.; a shilling for any weight 
over 5 Ib. and up to 8 Ib.; and so on. A straight line can 
be drawn through the corners of the figure, but this straight 
line does not pass through the origin. 

The mere plotting of a graph nowadays gives little trouble. 
Most modern books give instructions both simple and 
satisfactory. But a clear understanding of what has been 
done and a satisfactory interpretation of the completed 
graph often leave much to be desired. It is the interpretation 
that is the all-important thing. A graph is essentially a kind 
of picture, a picture to be understood. The pictorial element 
admits of a general interpretation simple enough for be- 



140 



CRAFTSMANSHIP IN MATHEMATICS 



ginners to understand, but as time goes on this interpretation 
must be made more and more exacting. 



The study of y = tnx -f c. Direct Proportion 

Experience convinces me that the study of the form 
y = mx should precede that of the form xy c. But pro- 
portionality of one kind or another underlies the whole thing, 
and the straight line and rectangular hyperbola should occupy 
a first place. 

Do not attempt to define for beginners the term function. 
The term should, however, be used from the first. " Here 
is an expression involving x, that is, a function of x." In 
time, drop the words " expression involving " and simply 
say " function of ". Let the word be used constantly; it 
will gradually sink in and become part of the boys' own 
mathematical vocabulary. 

Begin with a straight-line graph passing through the origin. 

(i) y = oc. What does this mean? That^y is always equal 
to x, i.e. that the ordinate is always equal to the abscissa, 



+3 



+2 



+I 



3 -2 




O +1 +2 
-I 

-2 
-3 



no matter what point on the line is taken, whether in the first 
or third quadrant. Thus in the figure we have the point 
(3, 3) in the first quadrant and the point (2, 2) in the 
third. 



GRAPHS 



141 



(ii)y == -oc. This is practically the same as before. The 
length of y is equal to the length of #, i.e. the length of the 
ordinate is equal to the length of the abscissa, but now the 
signs are different, whether a point is taken in the second 



-a; 




Fig. 25 

quadrant (as 3, 3) or in the fourth (as 2, 2). The graph 
runs from the left downwards, from the second to the fourth 
quadrant. 

(iii) 3y = 2x. This means that three times the length of 
tj|e ordinate is equal to twice the length of the abscissa. We 
may write, more simply, y = \x, and then we see that the 



(3,2) 




Fig. 26 



ordinate is always f of the abscissa. This is easily seen from 
any pair of values (save 0, 0) in a table: 

x = -.3 _2 +3 4-4 4 



y = _2 -1 



4-2 4-2} 



142 



CRAFTSMANSHIP IN MATHEMATICS 



No matter what point in the line is chosen, the ratio of (1) 
the -L r to the x axis to (2) the intercept on the x axis, i.e. 

the ratio ~, is always |. This ratio is constant; the triangles 

x 

formed by drawing perpendiculars are all similar. The slope 
of the line is always the same, i.e. the gradient of the graph 
is constant. 

(iv) 2y = 3x or y \x. Here the length of the 
ordinate is always 1^ times the length of the abscissa, but 

+ 3 
+ 2 

+ I 
-fl +2 +3 




-2 -/ 




the two are of opposite signs, as may be seen from any pair 
of values (save 0, 0). 

x = -2 -1 2 3 4 

y = 3 1J -3 -4J -6 

The graph runs from the left downwards, from the second 
to the fourth quadrant. 

Before proceeding further, give the class plenty of mental 
work from the squared blackboard, using a metre scale or 
a rod to represent the graph, holding it in various positions 
but always passing through some selected named point and 
through the origin, and asking the class to name the equations. 



GRAPHS 143 

I have known a class of thirty boys give almost instant 
response, one after the other, when tested in this way. 

See that the boys become thoroughly familiar with the 
difference between y = mx (same signs, slope from left 
upwards) and y = mx (opposite signs, slope from left down- 
wards). Also see that they are not caught by the alternative 
forms to these, viz. y mx = 0, y -f- mx = 0. 

The next step is to see that the boys understand the sig- 
nificance of the wi in the equation y = mx. They already 
know that when the coefficient of y is unity, the coefficient 

of x is a ratio representing ?-, i.e. the " steepness ", the 

x 

" slope ", or the " gradient " of the graph, and they are thus 
prepared for the general method of writing this ratio, viz. 
by the letter m. Do not begin with the general form m 9 and 
say that it represents the slope of the line, and then illustrate 
it with numerical examples. Begin with the numerical 
examples, in order that the boys may really understand the 
principle; then introduce the m as a sort of shorthand registra- 
tion of facts which they already know. 

The next step is to move the graph about parallel to 
itself, and to study the effect upon the written function; 
and so lead the boys to see that a graph which does not pass 
through the origin necessarily cuts off pieces (intercepts) 
from both axes (we neglect the case of a graph parallel to 
an axis). We may begin by graphing a few particular cases 
of the function y = mx > say y f # c: 

y = %x + 2 

y = 1* + 1 

y = t* 

y = f* - 1 

y = f* - 2. 

Show the pupils how to tabulate two or three pairs of values 
of each case, and how then to draw the graphs. They may 
then compare their results. 



144 



CRAFTSMANSHIP IN MATHEMATICS 



They will readily discover that the +2, +1, 0, 1, 2, 
represent merely the number of units the graph has been 
raised or lowered (the third case,;y #, being an old friend). 
The function proper, y = f#, is the same in all cases\ the 
slope is constant; the five lines are parallel. A perpendicular 
(ordinate) dropped from any point on the graph to the x axis 



\oc+2 



Fig. 28 

shows a right-angled triangle similar to all other similarly 
drawn triangles. In every case, the ratio of the sides round 
the right angle is given by the m, the coefficient of x. The 
number (the c) added or subtracted represents merely the 
bit of the y axis intercepted between the graph and the x axis. 
For this reason we call such bits of the y axis, intercepts. 

But when we raise or lower the graph above or below the 
origin, the graph really intercepts both axes. If the graph is 
raised above the origin, a portion of the y axis above the 
origin is intercepted, and a portion of the x axis to the left 



GRAPHS 



145 



of the origin, as well. If the graph is lowered below the origin, 
a portion of the y axis below the origin is intercepted, and 
a portion of the, x axis to the right of the origin, as well. 
How in each case are the two intercepts related? 

Consider the first of the above five expressions, viz. 
y = f # + 2. Instead of expressing y in terms of #, we may 
express x in terms of y, thus: 

v y = %x -f 2 

/. 3y = 2* -f 6 



Here the # intercept is 3, where we have precisely the 
same graph as before when the y intercept was +2. The 



-2 -' 



Fig. 29 



+ 1 




Fig. 30 



function is unaltered. So with the last of the five expressions, 
viz. y = f# 2. If we express x in terms of jy, we have 
x == 2^ + ^- The a? intercept is +3, and, as before, the y 
intercept is 2, the graph being identically the same. The 
function is unaltered, we have merely expressed it differently. 

Generally, however, we express y in terms of x, and the 
added or subtracted quantity (the c) represents a y intercept. 

The analogous results from the function y = f x i c 
may now be rapidly dealt with in the same way. 

Let the pupils occasionally check a graph by means 
of other pairs of tabulated values. For instance, from the 
function y -= f x + 2 we have: 



(K291) 



-3 +3 +6 +8 

+2 +4 +6 +7} 



11 



146 



CRAFTSMANSHIP IN MATHEMATICS 



Consider the last point (8, 7J), where OS = 8 and PS = 7J. 
The slope of the graph is determined by the sides round 
the right angle of any right-angled triangle determined 
in the manner aforementioned. In the main figure we 
see two such triangles (shown also as separate figures 




Fig. 31 

with the ordinates, x and j, in dark lines). The slope is 
determined 

PR ^ y _~- 2 = 7^ 2 = 2 

RQ x 8 3' 

- PS v 7* 2 

or by 



either by 



TS 3 + x 3 + 8 



- 2 



y _ 



Hence we may write either 
The two are identical. * ' 6 ' 6 + x ' 6 

Beginners are apt to confuse the value of m with the 
co-ordinates of some arbitrarily chosen point; e.g. to take 
the value (8, 7|) of the above point P, to convert it into the 

74 

fraction -?, and to call it m. It is a thing that wants watching, 
8 

The boys ought now to realize that, in y = mx + c, 
the c is of little consequence compared with the all-important 
m\ and that it may sometimes be convenient to ignore the 
c and to plot the graph in its fundamental form y = mx. 
Since it then passes through the origin, the function is more 
easily recognizable. 

The linear function should thus provide the boy with a 



GRAPHS 147 

preliminary training to enable him to see clearly how the 
relation between variables may be represented not only in 
equation form but pictorially. He should be able to discover 
the relation between the variables, that is, to discover the 
equation or law connecting them, to discover what function 
y is of x, to discover m. 

The beginner is often perplexed when told that Ax + By 
+ C = is the general form of a linear equation. Why 
those capital letters, he wonders. But if he first sees that 
his now familiar friend y = f x + -47- ma y be written 
3x + 4y = 18, he will understand that the new form provides 
a neater way of writing down the function, though the all- 
important m no longer reveals itself so readily. " When 
we write this new and neater form Ax -f~ By + C = 0, the 
only reason for using capital letters is that it enables us to 
identify it readily. Other forms and their specific uses you 
will learn all in good time. Why should we not have different 
ways of writing down the same function? May we not weigh 
up in the laboratory a piece of brass in ounces or in grams? 
Convenience dictates a choice of method." 

It is a good general plan to lead up to a general form 
through a few particular examples. To spring suddenly 
upon a class such a general form Ax + By + C = 0, before 
they have been suitably prepared, is not the sort of thing 
that an experienced teacher ever does. 

Independent and dependent variables are terms to be 
introduced gradually. Make quite clear that the x axis is 
always used for the quantity which is under our control and 
is quite " independent " of the other quantity, and that for 
this reason it is given the name independent variable; and 
that the y axis is used for values calculated from the formula, 
or for values observed in experiment, i.e. values which 
" depend " on the selected and controlled x values, and it is 
therefore called the dependent variable. Each time we change 
the value of our selected x quantity, calculation or observa- 
tion gives us a related y quantity; and the graph we draw is 
a picture .to show not only how these pairs of quantities are 



148 



CRAFTSMANSHIP IN MATHEMATICS 



related but to show that this relation is the same for every 
pair. 

Another way of expressing the connexion between the 
two variables is to say that the dependent variable is a function 
of the independent variable, the latter being often called the 
argument of the function, since we make it the basis of our 
argument. The graph of an equation shows how the function 
varies as the argument varies and is called the graph of the 
function; the abscissa is selected for the argument, and the 
ordinate thus represents the function. 

The Circle 

There is little to gain in spending much time over the 
circle, as it will rarely be used except to illustrate the solution 
of such simultaneous equations as x 2 + y 2 = 52, xy = 24. 
But it does serve to illustrate simply how a formula is affected 
where the graph is " pushed about ". We give the same 
circle in four different positions. 




Fig. 33 



Centre of circle at origin. Equation: x 2 + J> 2 = r 2 . 

The centre is pushed 1| units to the right; its co-ordinates 
are (1J, 0). The horizontal of the right-angled triangle is 
no longer x, but x diminished by 1J. 

Equation: (x 1 J) 8 + jy 2 = r 2 . 



GRAPHS 



149 



The centre is pushed 2j- units up; its co-ordinates are 
(0, 2). The vertical of the right-angled triangle is no longer 
jy, but y diminished by 2J. 

Equation: x 2 + (y - 21) 2 = r 2 . 




Fig. 34 



Fig. 35 



The centre is pushed li units to the left and 2J- units 
up. The horizontal of the right-angled triangle is x + 1|> 
and the vertical is y 2|. 

Equation: (* + H) 2 + (y 2) 2 - r 2 . 



The Study of ;ry = ^. Inverse Proportion 

The direct proportion graph we found to be a straight 
line. The inverse proportion graph (the rectangular hyperbola) 
is naturally the next for investigation. 

Let the learner himself plot some simple case: " 32 men 
take 1 day to mow the grass in the fields of a farm. How 
many days would it take 16, 8, 4, and 2 men, and 1 man to 
do it?" (An absurd example, really, but for our present 
purpose the weather conditions and the growth of the grass 
may be ignored.) 



150 



CRAFTSMANSHIP IN MATHEMATICS 



With half the number of men, twice the number of 

days would be required. 
With one- third the number of men, three times the 

number of days would be required. 

And so on. Hence, for graphing, we may write down these 
pairs of values. 



men \ 32 
days | 1 



16 
2 



8 
4 



2 
16 



1 
32 



The graph is evidently a smooth curve. Lead the class to 
discover that the product of each pair of values is constant, 
that xy is 32 in all cases. 




Fig. 36 



Now plot xy = k for several values of k, e.g. k 25, 
49, 64, 100, 225, 400, and examine the curves as a family. 
How are they related? 

1. A line bisecting the right angle at O divides all the 
curves symmetrically. 

2. The point where that line cuts the curve is the point 
nearest the origin; it is the " head " or vertex of the curve. 



GRAPHS 



151 



3. At a vertex V, x = y. Hence, . xy = k y x = y = 
. . in *jy = 25, the co-ordinates of the vertex V are (5, 5). 

4. Each curve approaches constantly nearer the axes, but 

k 
never reaches them. However great the length of x, y = - 

and y can therefore never be zero. Neither can x ever be 
zero. Either may be indefinitely small because the other 
may be indefinitely large, but neither can be absolutely zero. 
Hence we say that the axes are the asymptotes of the curve. 



oc at - 25 

d 
ecu 49 

DC of * 64 
ocu -;oo 




DC 



- 225 
-400 



/o 



/5 20 25 30 

Fig. 37 



This term means that the line and the curve approach each 
other more and more closely but never actually meet (asymptote 
= " not falling together "). 

5. The successive curves are really similar, although at 
first they do not appear so. But draw any two straight lines 
through the origin to cut the curves and examine the in- 
tercepted pieces of the curves (it is best to cover the parts 
of the figure outside these lines), and each outer bit of curve 
will be seen to be a photographic enlargement of the next 
inner bit. 

Boyle's Law is the commonest example of inverse pro- 
portion in physics. But the data (p and v) obtained from 
school experiments are usually too few to produce more 
than a small bit of curve, much too small for ready inter- 



152 CRAFTSMANSHIP IN MATHEMATICS 

pretation. But inasmuch as the law pv k seems to be 
suggested by the data, this may be verified in two ways: 
(1) find the product of p and v for each pair of related values 
and see if the product is constant; (2) convert the apparently 
inverse proportion into a case of direct proportion by plotting 

not v against p but - against p. The points thus obtained 
v 

ought to lie on a straight line, and the line may be tested by 
means of a ruler, or a piece of stretched cotton. Does the 
line pass through the origin? Why? 

There is probably little advantage in teaching boys to 
" push about " into new positions the rectangular hyperbola, 
though for purposes of illustration one or two examples may 
usefully be given. If the graph xy ~ 120, or y ^, is 
raised, say, 3 units, the function becomes y = -^..o + 3 or 
y 3 = ' jo. If it is i owere d 3 un i ts> y _j_ 3 ^ .1 .? o. If 

it is raised 3 units and then moved 4 units to the right, the 
function reads y 3 = i~ ( ] or y = ^ + 3. But the 
beginner is apt to find this a little confusing. It is best to 
let him keep the curve in a symmetrical position, and to 
continue to use the asymptotes for his co-ordinate axis. 

Negative values. Instruct the class to graph xy = 100 
for both positive and negative values. Then proceed in this 
way. 

11 When we plotted pairs of quantities from a linear 
function, we passed from negative values through the origin 
to positive values (or vice versa), and the graph was con- 
tinuous an unbroken straight line. Apparently, then, the 
rectangular hyperbola, though consisting of two separated 
parts, ought to be regarded as a single continuous curve. 
Is this possible? 

" The curve in the third quadrant is certainly an exact 
reproduction of that in the first. 

" Suppose the x axis indefinitely extended both ways, and 
a point Z far out to the right to travel along it towards O 
the origin. At any position it may be regarded as the foot 
of the ordinate of a corresponding point P on the curve. 



GRAPHS 



153 



As (for instance) Z x moves to Z 2 , P x moves round the curve 
to P 2 , and as ZO diminishes in length (ZjO to Z 2 O), the 
ordinate PZ increases (P^ to P 2 Z 2 ). But however long 
PZ may be, it gets still longer as Z gets still nearer O. In 
fact, it seems to become endlessly long, and yet we cannot 
say that the curve ever really meets the y axis, for it is absurd 













3U 

nc 


























Of) 


\ 








XI 

c 


/ ** 


oo 












15 




V 


PZ 




















ID 






\ 




























^ 


^^ 


Ff 


























r 


- 


-'< 




5 -2 

- 


p -/ 


5 -/ 


3 -: 


> O 


*-5 




r i 
2 


1 


5 2 


z , 2 


J i 






^ 


k 






-IO 






















\ 




^16 






















\ 




~9n 
























\ 


f 2S 
























\ 


---3o 















Fig. 38 

to speak of the quotient -$-. But if Z continues its march, 
it must eventually pass to the other side of O. And yet no 
interval can be specified to the left and right of O so short 
that there are no corresponding positions of P still nearer 
to the y axis on the right at an endless height and on the 
left at an endless depth. As Z proceeds along QJC, P simply 
repeats in reverse order along the curve in the third quadrant 
its previous adventure along the curve in the first. The 
crossing of Z over the y axis at O seems to have taken P 



154 CRAFTSMANSHIP IN MATHEMATICS 

instantaneously from an endless northern position to an 
endless southern position. We feel bound to regard the 
two curves as two branches of the same graph, for both are 
given by the function xy ~ k = 100. 

" If you plot xy k, the branches appear in the second 
and fourth quadrants." 

The above argument is always appreciated by A Sets, 
though naturally its implications are too difficult for them 
to understand until later. With lower Sets, it is futile to 
discuss the subject at all. 

With A Sets, too, the use of the term " hyperbolic func- 
tion " is quite legitimate. We called ax + b a linear function 
of x because the graph of y ax + b is a straight line. 
Similarly we may call any function that may be thrown into 

k 

the form + b a hyperbolic function of x, because the 

x ~\~ a 

k 

graph y = + b is a (rectangular) hyperbola. 

x -f- a 

The Study of y = x 2 . Parabolic Functions 

The pupil should master two or three new principles 
before he proceeds to the quadratic function. 

1. The first is the nature of a "root" of a simple equation. 
A very simple case will suffice to make the notion clear. 
The boy knows already that the root of the equation x 3 = 
is 3. Now let him graph the function y = x 3. 

Since y x 3 we have: 



* = 




- 


1 


2 


3 


y = X 3 = 


-3 


-2 


1 






The line crosses the x axis at +3, that is when y = 0, x = 3, 
and we say therefore that 3 is the " root " of the equation 
# 3 = 0. Of course we should never let a boy waste his 
time by actually solving an equation in this manner, but 
it serves to teach him that when the value of a function 



GRAPHS 



155 



equals 0, then the intercept on the x axis gives the root of 
the equation represented by the function. (Fig. 39.) 

The roots of related equations are easily derived. For 
instance, solve the equation #3=1. (Fig. 40.) 



'3.0 ' 




Fig. 39 



Fig. 40 



Write y = x - 3 = 1; i.e. y = x - 3, and y = 1. The 
graph of y = x 3 is the same as before; the graph of y = 1 
is a line parallel to the x axis, 1 unit above. The value of x 
in the equation # 3 = 1 is given by the intercept that 
y = x 3 makes with y = 1, i.e. 4. In other words the root 
of the equation is the x value of the point of intersection of 
the two lines. 

Evidently we have the clue for solving graphically two 
" simultaneous " equations, say, x2y=l, and 2*+3y=16. 
The lines cross each other at the point P (5, 2). This pair of 
values satisfies both equations (let class verify). A line drawn 
through this point parallel to the x axis is y = 2. Hence 
the value of x for both lines where they cross y = 2 is 5. 
The 5 represents the intercept on the line y = 2, made by 
each of the given lines. (Fig. 41.) 

2. A second preliminary principle to be mastered concerns 
the method of making out tables of values for graphing. 
Having decided what values of x are to be used (this is a 
question of experience), write them down in a row, then 



156 



CRAFTSMANSHIP IN MATHEMATICS 



evaluate the successive parts of the function, one complete 
row at a time. The mental work proceeds much more easily 
this way than when columns are completed one at a time. 
For the sake of comparison, we will set out selected values 



12 

Px 



X -2i 

2oc + 5i 

L 

Fig. 



1 

IG 



of the function 4# 2 4# Je^, in two ways, one by addition, 
one by multiplication. Show the learner why the results are 
necessarily identical. 



x= 


o 

36 
12 


-2 

16 

8 


-U 


-1 

4 
4 





i 

1 
-2 


+ 1 

4 
-4 


+ 2 


+ 2i 


+ 3 

36 
-12 


+4 

64 
-16 

33 

i 


4r 3 -: 
-4z- 


9 
6 






16 

-8 


25 

-10 


y~4:X z 4:X~ 15 -- = 


33 


9 





n 


-15 


-16 


-15 


-7 





9 



Since the function factorizes into (2x + 3) (2x 5), we 
may set out the values of the factors and multiply, instead 
of adding as before: 



x~ 


-3 


-2 


-li 


-1 





i 


+ 1 


+ 


+ 2J 


+ 3 


+4 


(2*+3) = 
(2*-5) = 


-3 
--11 


-1 

g 




o 


+1 

-7 


3 
-5 


4 
_4 


5 
-3 


7 
-1 


8 



9 
1 


11 
3 


y = 4#* 4# 1 5 = 


33 


9 





-7 


-16 


-16 


-16 


-7 





9 


33 



GRAPHS 



157 



3. A third preliminary principle concerns scales. Different 
scales for the two axes are often desirable, though in the early 
stages of graphing different scales are not advisable. The 
learner should recognize the normal slope of the straight 
line and the normal shape of the curve. Only in this way 
can he recognize and analyse the purely geometrical properties 
of the graph. But with the study of the parabolic function, 
if not before, the " spread " of the numbers should be taken 
into account. Moreover, a good " spread " to the parabola 
is an advantage, in order to obtain accurate readings of the 
x intercepts. 

We now come to the actual graphing of the function. 



2,4 



-u-4 



2.4 C 



2 o 



-2 



- X 



X 



Fig. 42 



Let the boy be first made familiar with the graph of 
the normal function y x 2 , the parabola being head down 
and the co-ordinates of its head (vertex) being (0, 0). Let 
him see that the curve cuts any parallel to the x axis in two 
points, e.g. 1 and 1, \/2 and \/2, &c. The curve is 
symmetrical with respect to the y axis. Note that, with the 
same scale for both axes, there is not much spread to the curve. 

Now we will graph the function y 4# 2 4^15, 
taking the sets of values for x and y from either of the tables 
on the previous page. To obtain a greater " spread ", we 
adopt a larger scale for the x axis. The curve cuts the x axis 
(when . . y = 0) in two points, viz. 1 and 2 (these values 



158 



CRAFTSMANSHIP IN MATHEMATICS 



are also seen in the tables), and these are therefore the roots 
of the equation 4# 2 4# - 15 0. 

From the same graph we may obtain the roots of the 
equations 4# 2 4x 15 = 9, or 4# 2 4# 15 7, or 
4# 2 4# 15 = z y where z = any number whatsoever. It 
is simply a question of drawing across the curve a parallel 



V 



-3 



\ 



33 



\ 



-2 



ttLfi) 



* 



30 



10 



+ 2 



/v 



K33) 



Fig. 43 



to the # axis, and of reading the values of x from the points 
of intersection. For instance, if 4# 2 4# 15 = 9, the 
parallel to be drawn is x = 9, and this cuts the curve in 
x = 2 and 3, which are therefore the roots of the equation. 
These values of x may, of course, be seen from our tables 
where y 4# 2 4# 15 = 9, but they are easily estimated 
from the graph itself, if this is reasonably accurate. 

A function may sometimes be conveniently divided into 
two parts, and each part treated as a separate function and 



GRAPHS 



159 



graphed. The intersection of the two graphs will then give 
the roots of the equation. Really we have two simultaneous 
equations; e.g. 

if 4* a - 4* - 15 = 0, 

then 4* 2 = 4* + 15. 



Hence we may write 
and 



y = 4# 2 

y = 4* + 15 



The line cuts the curve at the points x = 1 and 2, and 
these are the roots of the equation 4# 2 4# 15 = 0, as 






10- 



** 






*0 



-ocr 



-2 -I 



- 4x-M5 

Fig. 44 



before. It should be noticed that this last figure does not 
represent the graph of the function y = 4# 2 4# 15, 
though this graph is now easily drawn by superposing the 
4* 2 graph on the 4* + 15 graph. If Y t = 4* 2 , Y 2 = 4^: + 15, 
and Y = 4^ 2 - 4* 15, then Y = Y x - Y 2 . Hence any 
ordinate of Y may be obtained by taking the algebraic difference 
of the corresponding ordinates of Y! and Y 2 . Let the pupils 
draw the Y graph from their Y x and Y 2 graphs, and verify. 



160 



CRAFTSMANSHIP IN MATHEMATICS 



The function might have been broken up in another way 

If 4x a - 4* - 15 = 0, 

then 4# a 4* 15. 

Hence we may write y = 4jc 2 4# 

and y = 15. 



(-3,48) 



\ 


























/ 


K48) 


\ 


























/ 


\ 
























, 


y 




\ 






















rJ 






\ 


























(- 


'^' 


\ 


















i 


b,2 


> 






\ 










If 


= if 






/ 










/ 


\ 








^ 






/ 


\ 






,l 


tf 


( 


l,8> 


\ 










^ 


/ 

'2,8) 




^ 


15) 










S 


X 






/ 










5-2-10 


(^ 


1) 1 2 3 4^ 




Fig. 45 



Here are the graphs of these two functions. The latter cuts 
the former at # 1^ and 2J, the same roots as before. 
The easiest way to discover where the parabola 4# 2 4# 15 
crosses the x axis is to express the quadratic function as a 
product of two linear functions, viz. (2# + 3) (2# 5) = 0. 
Hence either 2# + 3 = or 2# 5 = 0, i.e. x= f or -ff. 
Thus from the two linear functions we form two simple 
equations, the roots of which are the roots of the quadratic 
equation. 



GRAPHS 



161 



We will plot these two linear functions (see the second 
table, p. 156). (The lines happen to be parallel. Why?) 
The graph of the quadratic function is readily obtained by 
multiplying together corresponding y values (again refer to 
second table, p. 156). For instance, at 2 the y value of 
2x + 3 is 1 and the y value of 2x 5 is 9. The product 
of 1 and 9 is +9. Hence at -2 the y value of the 
quadratic function is +9, i.e. the point (2, +9) is a point 




Fig. 46 

on the curve. By pursuing this plan we may obtain fig. 43 
over again. 

The boy ought now to realize that he may graph his 
function in a variety of ways. But do not encourage him to 
think that the normal process of solving a quadratic equation 
is to graph the function. Not at all. The important thing 
for the boy to understand is that every algebraic function 
can be thrown into a picture and that this picture tells a 
story. What the algebra means to the geometry and what 
the geometry means to the algebra are the things that matter. 
We are dealing with the same thing, though in two different 
ways, and the closeness of the relationship should be seen 
clearly. As with the linear function, so with the parabolic 

(B291) I* 



162 



CRAFTSMANSHIP IN MATHEMATICS 



function: the boy must see the result of " pushing the graph 
about ". 



If 

then 
or 



4x 15 = y, 
2 - 4* + 1 = y + 16, 
(2x - I) 2 = y + 16. 



If we compare this with the normal form x 2 = y> we see that: 

2x I has taken the place of x 
and y + 16 has taken the place of y t 

i.e. instead of x = 0, 2x I = 0, or x = , 

and instead of y = 0, y -f 16 = 0, or y = 16, 

i.e. the head of the parabola is not (0, 0) but (J, 16) as in 
fig. 43. Clearly the graph of 4# 2 4# 15 is identical 




- oc 



3oc 



GRAPHS 163 

with the graph of 4# 2 , except that it has been pushed J unit 
to the right, and 16 units down. (The scale difference must, 
of course, be borne in mind.) 

This identification of similar functions is of great im- 
portance throughout the whole range of algebra. One of 
the greatest difficulties of beginners is to see how the form 
of a normal function may be obscured by mere intercept 
values. 

Family of parabolas. Let the boy graph a few related 
parabolas like the following: y = x 2 \ y = 2# 2 ; y 3# 2 ; &c. 
For 2# 2 , the ordinates of x 2 are doubled; for 3# 2 , tripled; 
and so on. Grouping of this kind helps to impress on the 
learner's mind the relationship of the curves. 

A metal rod bent into the shape of a parabola, with an in- 
conspicuous cross-piece for maintaining its shape and for mov- 
ing it about the blackboard, is useful for oral work in class. 

Contrast the parabola y ax 2 + bx + c when a is 
negative with that when a is positive. With a negative, the 
curve is " head up "; e.g. 7 + 3# 4# 2 gives such a parabola. 
Fig. 49 shows another. Give the boys a little practice in 
drawing parabolas in this position. They should also draw 
one or two of the type x = y 2 and x = y 2 , and carefully 
note the positions with respect to the axis. 

Turning -Points. Maximum and Minimum Values 

The pupil has learnt that in y = 4# 2 4# 15, the head 
of the parabola is (J, 16). He sees that the equation 
4# 2 4# 15 has two roots whenever y is greater than 
16. For example if y = 9, x = 2 and 3; if y = 7, the 
roots are 1 and 2; if y = 15, the roots are and +1. 
But if y = 16 the two roots are equal, each being *5. ' If a 
line parallel to the x axis is down below y = 16, it does not 
cut the curve at all, so that if y is less than 16, x has no 
values, or, as is generally said, " the equation has no roots ". 
For instance, if we give y the value 17, and work out the 
equation 4# 2 4# 15 = 17 in the ordinary way, we 



164 



CRAFTSMANSHIP IN MATHEMATICS 



find that x = 



V-i 



But these values of x have 



no reality because we cannot have the square root of a negative 
number. The graph tells the true story. Instead of saying 
that the equation has two unreal or " imaginary " roots, 
we may more correctly say that, when the value of y is less 
than 16, x has no value at all, simply because the y line 



" -7 



Fig. 48 

does not now cut the curve at all. The y line is " out of the 
picture ". 

As a point moves along the curve from the left downwards, 
the ordinate of the point decreases until it reaches the value 
-16, then a turn upwards is made, and the ordinate begins 
to increase as it ascends to the right. The point (+-5, 16) 
is the turning-point of the graph, and the value 16 of the 
ordinate is called the turning value of the ordinate (or of the 
function). That value of the ordinate is its minimum value. 
If the graph was one with its head upwards, the turning- 
point would be at the top and would be a maximum value 
(see fig. 49). 



GRAPHS 



165 



Thus the pupil must understand clearly that, in the case 
of any parabolic function, the head (vertex) represents a 
kind of limiting value of y. Each value of y corresponds to 
two different values of x, though the head of the curve seems 
to be an exception. Strictly speaking, the head corresponds 
to only one value of #, but it is convenient to adopt the con- 
vention that x has in this case two identical values. Beyond 
the head, outside the curve, x can have no values. Some 
quadratic equations have two roots, some OPP (two identical), 
some none. Do not talk of " imagi.iary ' roots: that is 
nonsensical. We shall refer to this point again, in the chapter 
on complex numbers (see Chap. XXVII). 

The pupil should note how slowly the length of the 
ordinate changes near the turning-point of a parabola. In 
fact this characteristic of slow change near a turning-point 
is characteristic of turning-points in all ordinary graphs. 
Let the pupil plot on a fairly large scale y = x 2 for small 
values of x. 




no 










x- 


^x. 










p 




I 


2 / 


? 


4 


^s 
5 


X 


7 







-10 




/ 


^ 










\ 






-to 
















\ 




/ 
















s 


\ 


-V 


f 


















\ 


1 




















\ 



<-(2x 2 -!9oc+35) 

Fig- 49 



Show the pupil how the graph tells him at a glance 
where the values of y (the function) are positive, say for 
y = 19x 2x 2 35. The part of the curve above the x axis 



166 CRAFTSMANSHIP IN MATHEMATICS 

corresponds to values of x between 2 and 7. But the values of 
y (= 19# 2x 2 35) above the x axis are positive. Hence 
the expression 19# 2x 2 35 is positive between the values 
2 and 7. If any values outside these are tested algebraically, 
the expression is seen to be negative. (Fig. 49.) 

It may be emphasized again that quadratic equations 
should be looked upon as merely one interesting and useful 
feature in the general elementary theory of parabolic functions. 
Do not forget practical applications of the parabolic function; 
e.g. falling bodies in mechanics. 



Simultaneous Equations 

Practice in solving various types of simultaneous equations 
should be given less with the idea of finding the actual roots 
of the equations than for the purpose of studying the relative 
positions and the intersections of the graphs. We will refer 
briefly to two typical examples. 

1. Consider the equations: 

x z + y* = 97 \ and x* + y* = 20 



= 97 \ 
= 36 / 



xy = 36 and xy = 36 

# 2 + y 2 = 97 is a circle with its centre at the origin and 
radius A/97; and xy 30 is a rectangular hyperbola symmetri- 
cally placed in the first and third quadrants, with its vertices 
at a distance of V2 X 36 from the origin. As V2 X 36 is 
less than V97, the circle cuts the hyperbola in four points, 
symmetrically placed. In the second case, since V20 is less 
than V2 X 36, the circle does not cut the hyperbola, and 
there are no roots. (Fig. 50.) 
2. Consider the equations: 



X = \r 



Here we have two parabolas, one with its apex downwards, 
touching the axis of x, two units to the right of the origin^ 



GRAPHS 



167 









d 










, 








y 


\ 












^ 






\^ 
















\ 








/ 








/* 


\ 










s^ 


^ 


/ 


X 












/ 




\ 








/ 




/ 


\ 












o 






1 


A 






\ 


O 




/ 








\s 


S 








/ 






-^ 


\ 


^~ 


^ 










" 


A 






s 










\ 












































3C% o/ - 97 
ocy =36 
OV =A/^36; OA - V97 


x'H-^-20 

OC'U' - 3G 
OA - ~/ZO ; OV- v'Z x l 



Fig. 50 



the other symmetrically astride the x axis, with its apex at 1 
to the left. The roots are readily obtained approximately by 
measurement of the co-ordinates of the intersections. (Fig. 51.) 

jc 



4 - 



3 (- 



fo.O 



(c5 r 

-2 






(20) 



-2) 



- .( X - 2) 



Fig. 51 



168 



CRAFTSMANSHIP IN MATHEMATICS 



Higher Equations 

The pupils should study a few cubics graphically, if only 
that they may gain confidence in a method of general applica- 
tion. 

The normal form of the cubic (y = # 3 ) is easily graphed 
and remembered. 




- re 

Fig. 52 

Consider the equation S(x 1) (x 2) (x 4) = 0. 
Let 8(* - 1) (x - 2) (x - 4) - y. 



* = 





1 


U 


2 


3 


4 


5 


(*-!) = 


-1 





i 


1 


2 


3 


4 


(* - 2) = 


2 


1 


i 

2 





1 


2 


3 


(* - 4) = 


4. 


-3 


-21 


-2 


j 





1 


y = 8(* - 1) ( - 2) (* - 4) = 


-64 





5 





-16 





96 



The curve cuts the x axis at points 1, 2, and 4, which are 
therefore the roots of the equation (as, of course, we know 
at once from the factors). (Fig. 53.) 



GRAPHS 



169 




(5 -16) 



^ = 8(x-l)(x-2)(x-4) 
Fig. 53 

Now consider the equation # 3 7# + 4 = 0. 

Since * 7* + 4 = 0; /. * 3 -= lx 4. 

Let ^ = Y I; 7* - 4 = Y 2 ; Y 3 - Y t - Y 2 - ^ 3 - lx + 4. 

We will tabulate values for Y a , Y 2 , and Y 3 . 







2 


-2 


-1 





+ 1 


+ 2 


+ 3 


Y, = 


x 3 = 


-27 


-8 


j 





+ 1 


+ 8 


+ 27 


Y 2 = 


lx - 4 = 


-25 


-18 


-11 


-4 


+ 3 


+ 10 


+ 17 


Y 3 - 


^ 3 - 7 -i- 4 = 


-2 


-1-10 


+ 10 


+ 4 


-2 


2 


+ 10 



We will now plot Y x (= # 3 ), a normal cubic, and Y 2 
(= 7# 4) a straight line, and so solve the equation. The 
latter cuts the former in three points, viz. where x = 2-90, 
60, 2'29, which are therefore the three roots. But the 



170 



CRAFTSMANSHIP IN MATHEMATICS 



figure (fig. 54) does not show the graph of y = of - 7x + 4, 
the original function. To draw this graph, we may either use 
the values of Y 3 in the table, or superimpose the above two 
graphs, Yj and Y 2 , on each other, remembering that 




7 



OC 

7oc-4 

Fig. 54 



Y 3 = Y t Y 2 , and that therefore we may obtain any ordinate 
for Y 3 by taking the difference of the corresponding ordinates 
for Y! and Y 2 . For instance, the ordinate at x = 2 is 8 
for Y! - a, and -18 for Y 2 = 7x - 4, and for Y 3 (= Yj-Yg) 
is therefore -8+ 18, or + 10. And so generally. This 




y 



. 55 



time the roots of the equations are given by the intersection 
of the curve with the x axis, the values (2-90, -60, 2-29) 
being, of course, the same as before. (Fig. 55.) 



172 



CRAFTSMANSHIP IN MATHEMATICS 



The Logarithmic Curve 

We dealt with the A B C of Logarithms in Chapter XI, 
and we now come to the logarithmic curve, the use of which 
is, of course, not as a substitute for the tables but as a justifi- 
cation of the extension of the laws of indices from positive 
integers to fractional and negative values. The boy has to 
learn, too, that the curve is really a picture of a small set of 
tables. He should therefore be taught to plot a curve from 
first principles, and to use it as far as he can. 

Let him first become familiar with the general form 
of the curve. For instance he might plot y = 2 X , 3*, 5*. 



Go 
50 
40 
30 
20 
IO 




OC 











1 




















1 








1 










/ 








x 


' 






' 


s 







- 2 



l*-*2 



Ex 

FiR. 56 



3 4 



- 3 

4 -8l 




T 



Ex. 5 - 125 



Show the advantage of changes of scales. Draw two or three 
extended logarithmic curves on the blackboard, and spend 
a few minutes in oral work, e.g. 2 8 ? 3 6 ? 5 4 ? (approximate 
answers are of course, all that can be expected). 

The next step is to deal with the evaluation of fractional 
indices in y 10*. Let the class graph y 10* up to x = 3, 
on a fairly large scale, drawing the graph from the integral 
values x = 1, 2, 3. " If the index law holds good, we ought 
to be able to obtain by readings from the graph such values 
as IO 1 * and IO 2 *. But our graph is necessarily very rough; 



GRAPHS 



173 



we had such a few points with which to plot it. We must 
try to construct a better curve. 

" Let us use our arithmetic for constructing the curve, 
say a curve representing values from 10 to 10 1 . The more 
values we find, the more points we shall have for plotting 
our curve. How many? Say 7 between 10 and 10 1 , viz. 

10s 10*, 10&, 10*, 10s 10s 10&." 



Begin with 10* = 10 > = 
Then 10* - 10'. = 
Then 10s = 



3-162. 
1-779. 
= 1-333. 

We have 4 more to find, viz. 

10s 10, 10s 101. 

I0l = (10<) 3 = (1-333) 3 - 2-371. 

10 == (10<)> - (1-333)-' = (1-333) 2 X (1-333) 3 
- 1-779 X 2-371 = 4-217. 

I0l = 10? - (101) 3 - (1-779) 3 = 5-623. 

105 = (lO 1 ^) 7 - (1-333) 7 - (1-333) 4 x (1-333) 3 
= 3-162 x 2-371 = 7-497. 

If the arithmetic is distributed amongst the class, it is quickly 
done; very little explanation is necessary, provided previous 
elementary work in powers and roots was understood. 

Now the boys can make up their table of values, changing 
the vulgar fractions into decimal fractions; then plot their 
points, and draw the curve. 



X = 





125 


25 


375 


5 


625 


75 
5-623 


875 


1 

10 


y = 10* = 


1 


1-333 


1-779 


2-371 


3-162 


4-217 


7-497 



The class may now be given a few multiplication and division 
sums to work, for the purpose of checking their curve. (Of 
course they cannot read to more than 2 places of decimals.) 



174 



CRAFTSMANSHIP IN MATHEMATICS 



1. Multiply 3-79 x 2-38. 

From the graph, 3-79 = 10' 6 ' 9 and 2-38 = 10' 876 . 

/. 3-79 x 2-38 = 10'" x 10-" = 10' 9 = 9-02 (from the graph). 

Now verify by actual multiplication. 



*90 



JE 
3 



* 

422 



i2S~ 

2L_ 

1-78 



E%- 



10 



^5" 



UC 



7&: 



_LJ-A 



^Q* 55 . 



02 



X' 



625 



ROWERS TO BASE 10. 
Fit?. 57 

2. Divide 9-02 by 2-38. 
9-02 + 2-38 

= 10-9" .i. XO-876 = JQ.955-370 

Now verify by actual division 



Now let the class write into their graph, by interpolation, 
the index values of the integral numbers 1 to 10. (Some 



GRAPHS 



175 



teachers make the boys learn off these values to 3 places of 
decimals.) The boys' interpolations resulting from their 
own measurements will necessarily be very rough and at 
this stage a prepared graph of the following kind might be 
given them. 



X = 





301 


477 


602 


699 


778 


845 


903 


954 


1-00 


y = 10* = 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 

















































































/ 








9 


-10 


954 




























1 






































/ 










ft 


= 10 


903 


























/ 






































/ 












7 


-10 


645 
























/ 






































/I 














G 


-10 


776 




















./ 


' \ 






































r 
















5 


-10 


699 


















/ 






































/^ 




















4 


-10 


602 














/ 






































/^ 
























3 


-10 


477 








, 


X 




































/> 


^ 




























2 


-10 


301 


^ 


<^ 




































^ 






























^ 


, 


-*" 


































































i 




















50 








fA 




6C 




-69 




t 

77JS 


\ 

an 


$ 


3 *9( 


t 


, 







i 




'4 




t 


f 




$ 


C 


> 


V 




i 


i 




1 





POWERS TO BASE I0 f> 
Fig. 58 



The term " logarithm " may now be introduced. " It 
is just another name for index." Set out a multiplication 



176 CRAFTSMANSHIP IN MATHEMATICS 

sum in parallel, showing the related methods. Emphasize 
the fact that the two things are the same, except in appear- 
ance. 

Multiply 4-73 by 1-84. 



1. 

Let 4-73 x 1-84 = x. 
x = (4-73 X 1-84) 
== 10 fl76 X 10 265 (graph) 



= IO 940 ; 
/. x = 8-70 (graph). 



2. 

Let 4-73 x 1-84 = x. 

log x = log (4-73 X 1-84) 
= log 4-73 -f log 1-84 
= -075 -f -265 (graph) 
= -940; 
.'. x = 8-70 (graph). 



Now give the boys just one page of 4-figure logarithms, make 
them work out a few examples in both ways, and see they 
understand that the two ways represent exactly the same thing. 
It ought now to be possible for the boys to proceed with 
logarithms in the usual way, and really to understand what 
they are doing. 

Graphs and the " Method of Differences " 

The nature of a graph may easily be investigated by means 
of the method of differences. A series of equidistant ordinates 
is drawn, beginning at any point on the graph. The heights 
of the ordinates are measured, and a table is made of the 
first, second, third, . . . differences. If the graph is a straight 
line, the first difference will be constant; if a parabola, the 
second difference; if a function of the third degree, the third 
difference; and so on. Hence by examining the differences of 
the ordinates, we can determine the degree of the function 
which corresponds to the graph. This is a useful principle 
for the boys to know. 

Books to consult: 

1. Graph Book, Durell and Siddons. 

2. Graphs, Gibson. 



ALGEBRAIC MANIPULATION 177 

CHAPTER XVIII 

Algebraic Manipulation 

Common -form Factors 

During the last 30 years there has been amongst the older 
boys of schools a serious falling off in their power of algebraic 
manipulation. Nowadays, there is often a sad lack of easy 
familiarity with even the simpler transformations in algebra 
and trigonometry. Although a great deal of bookwork is 
done and mastered, the valuable old transformation exercises 
receive too little attention, with the result that there is often 
a good deal of uncertainty about everyday working algebraic 
procedure. 

Readiness in manipulation is the key to algebraic success. 
Pupils must acquire facility in the manipulation of common 
algebraic expressions. 

The factors to be mastered in the first year of algebra are 
few, but they are of fundamental importance and must be 
taught thoroughly. In the early stages they should be associated 
with arithmetic and geometry, if only in order that the 
pupils may be convinced of their usefulness. 

The early forms are, 

ab ac = a(b c), 

and a 2 - b 2 = (a + b) (a - 6); 

and the expansions (a + b) z = a 2 + 2ab -f b 2 
and perhaps (a &) 3 - a 3 + 3a*b -f 3ab 2 + fc 3 . 

Let factors be first looked upon as a device for simplifying 
formulae, and for putting these into shape for arithmetical sub- 
stitution. It is a good plan to begin with obvious geometrical 
relations and base upon these an algebraic identity. But do 
not talk of " proving " the truth of the geometrical pro- 
position. The illustrations in Chapter XVI, pp. 134-7 
typify the kind of thing to be done. 

(B291) 13 



178 



CRAFTSMANSHIP IN MATHEMATICS 



The elementary standard forms (a + b) 2 , (a A) 2 , 
a 2 b 2 y being well known, verified by a few numerical 
examples, and illustrated geometrically, a first element of 
complexity may be introduced into them. 

The a and the b may be regarded, respectively, as, say, 
a square and a circular box, into each of which we may put 
any algebraic expression we please. Thus we may write: 

D 2 - O 2 = (D + 0)(D - O), 
and then fill up, say with p 2 and q 2 respectively, in this way: 




.e. 



- tf = (p* 



Q) (/>- 



Such a device is very useful, but do not carry such an ex- 
tension very far at first. Wait a year, and then with harder 
examples push the principle home. 

The expansions (a b) 3 are probably best postponed 
until the second year, though when they are taken up they 
should be associated with a geometrical model. A 6-in. or 8-in. 
cubical block, sawn through by cuts parallel to each pair of 
parallel faces, makes a suitable model, and may be prepared 
in the manual instruction room. Or a cube cut from a bar 
of soap may be used, if a very thin-bladed knife is available 
for cutting the sections. We deal first with the identity 
(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 . If each edge of the 
cube is cut into two parts a and i, the original edge being 
a + b, and a being > 6, the cut-up cube evidently consists 
of eight portions, viz. a larger cube # 3 , three square slabs 
of area a 2 and thickness b, three square prisms of length 
a and square section i 2 , and a smaller cube b 3 . 

Then the class sees at once that 

(a + 6) 8 = a 8 + 3a*b + 3a& 8 + b*. 



ALGEBRAIC MANIPULATION 



179 



But they should discover this identity from the model for 
themselves, and not be told. 

The same model may be used for the identity 



but the manipulation is a little more troublesome. The 
whole composite cube must now be called # 3 , and the thick- 
ness of the other seven parts (slabs, prisms, and small cube) 
should be called b. The larger of the two cubes within the 
whole composite block is evidently (a A) 3 . When actually 
handling the model it is easy to see that this cube (a 6) 3 
with the three slabs (3a 2 b) and the little cube (6 3 ) are together 
equal to the whole composite block (# 3 ) plus the three prisms 
(36 2 ), i.e. 

(a - b)^ + 3a*b + 6 3 - <z 3 -f 3ab\ 
or (a - b)^ - 3 - Sa*b -f Sab* - b*. 

The three slabs " overlap ", a fact which tends to perplex 
most pupils. 

It is really better to cut up two cubes and have two 
models, one to be kept in its eight separate pieces, the other 





Fig. 59 



to be glued up again without the cube (a i) 3 and looking 
something like three of the six sides of a cubical box. Unless 
the teacher is pretty deft in manipulating such a model, it 
had better not be used, or the class will get more amusement 



180 CRAFTSMANSHIP IN MATHEMATICS 

than instruction from his efforts. It is obvious that since 
a AB, the second model (the three-sided shell) is less than 
the three slabs 3a 2 6 by the three prisms Sab 2 diminished by 
the little cube i 3 . 

I.e. shell - 3a 2 b - (3ab 2 - 6 3 ) 



add the removed cube (a 6) 3 to each side: 

shell + (a - 6) 3 - (a - b)* + 3a*b - 3ab 2 + b*, 
i.e. a 3 = (a - b)* + 3a 2 6 - 3ab* -f- 6 3 , 

i.e. (a - b)* - a 3 - 3a 2 6 + 3a6 2 - 6 3 (as before). 

This on paper looks complicated. With the model in the hand 
it may be made clear at once. The case seems complicated 
because what we have called a slab a 2 b consists of four pieces 
of wood, each of the thickness b, viz. a slab (a b) 2 in area, 
two square prisms each (a b) long, and a cube i 3 . 

The boys always look upon it as a pretty little puzzle. 
Let them build up the cube a 3 themselves, beginning with 
the cube (a 6) 3 , and adding and subtracting the other 
pieces one by one. The whole difficulty comes about from 
calling the edge of the whole cube a as compared with the 
previous example when a referred to part of the edge. 

A further identity for the boys to discover from their 
model is: 



The whole cube may be called a 3 and the removable cube 
i 3 . Lay out the seven pieces, all of thickness (a 6), on the 
table. The united area obviously is: 

3ab + (a - 6) a 
= a 2 + ab + b 2 ; 

:. volume = (a b) (a 2 + ab + b*) 9 
i.e. a 3 - & 3 = (a - b) (a 2 -f ab + 6 2 ). 

Verify all these identities by a variety of numerical calculations, 
and so emphasize the utility of the alternative forms. 

It is a curious fact that Form IV boys are prone to forget 



ALGEBRAIC MANIPULATION 181 

the factors of a 4 -f a?b 2 + A 4 . It is a good thing to ask them 
occasionally for the factors of (a Q A 6 ). They will give 
them readily enough: 

= (a 8 + 6 3 ) (a 3 - & 3 ) 

= (a + b) (a 2 - ab + b 2 ) (a - b) (a 2 + ab + b 2 ). 

Now ask them to multiply the four factors together again, 
in pairs: 

(a -f b) (a b) a 2 b 2 (readily given), 

(a 2 ab + b 2 ) (a 2 + ab + b 2 ) (generally forgotten). 

If the product is not forthcoming, ask for the factors of 
# 4 + a 2 b 2 -|~ 4 and give them the hint of adding and sub- 
tracting a 2 6 2 , thus: 

a* + a 2 b 2 + 6 4 
= ( 4 -f 2a 2 b 2 + 6 4 ) - a 2 b* 

- (a 2 -f b 2 ) 2 - W 

- (a 2 + ab + b 2 ) (a 2 - ab + b 2 ). 

Come back to this twice a term, until it is known. 



Algebraic Phraseology 

Each successive school year will demand its quota of 
further manipulative work until in the Upper Fifth, especially 
the top Set, the boys become expert. The four or five years' 
course of instruction must be organized in such a way that 
the difficulties of manipulation are carefully graded. Im- 
press on the boys that ready manipulation is the key to success 
in the greater part of algebra and therefore to the greater 
part of trigonometry, conies, and the calculus. 

Let your phraseology be accurate, and use it consistently, 
exercise after exercise, lesson after lesson, and see that the 
boys gradually acquire the use of phraseology of the same 
degree of accuracy. 

" Jones, what is the first thing to do?" " Rearrange the 
terms." 



182 CRAFTSMANSHIP IN MATHEMATICS 

" How?" " Write down all the plus terms first, and then 
all the minus terms." 

" Then?" " Put the plus sign . . ." 

" No. That is not the way we decided to say things." 
" Add up all the plus terms and write down the sum, pre- 
fixed by a plus sign; then add up all the minus terms and 
write down the sum, prefixed by a minus sign." 

" Smith: lastly?" " Take the difference between the 
two sums, and prefix the sign of the larger." 

Remember the slow boys and the amount of practice they 
need until the soaking in is complete. Then all is well. 

There are certain common algebraic terms which, though 
of fundamental importance, are often loosely used. Formal 
definitions to be learnt by rote are unnecessary, but con- 
sistently accurate usage should be adopted from the outset. 
Introduce the terms one at a time and make each new one 
part of the everyday jargon of each lesson for a few weeks. 
We refer to such terms as mononomial, binomial, degree and 
dimensions , homogeneity and symmetry, and so forth. 

" In algebra, a letter, or a product of two or more letters, 
or of letters and numbers, in which there is no addition or 
subtraction, is called a term, or a mononomial, e.g. #, # 2 , x 2 y, 



" If the same letter occurs more than once in a term 
we write the letter down once, and at the top right-hand 
corner we write a figure to show the number of times it 

occurs, e.g. xxx is written # 3 , aaaa is written 4 . 

i q 

" A term may be integral^ as ab 2 ; or fractional, as . 

x 

" The degree or the dimension of a term is the sum of the 
indices of the named letters; e.g. the term x 2 y 3 is a term of 
the fifth degree, or a term of five dimensions. 

" A binomial consists of two terms connected by the 
sign + -or ; a trinomial of three terms; a polynomial of 
more than three." 

All this is just the stock phraseology of the classroom. But 
let it be carefully thought out and consistently used, in order 



ALGEBRAIC MANIPULATION 183 

that the boy may soon get to know the precise significance 
of the new vocabulary. 

We have already referred to the term function. Use it 
consistently and use it often. 

Such a term as the law of commutation is hardly worth 
mentioning at all unless it be in Form VI, where algebraic 
theory is being minutely discussed. The boys will know 
from their arithmetic that the mere order in which terms are 
arranged for addition purposes is immaterial. So with 
multiplication: the notion of commutation is imbibed with 
the multiplication table; 5 sevens gives the same product as 
7 fives. Thus, any elaborate formal explanation that 
d+c + a + b a -{- b -{- c -{- d, or that b 2 ac is the same 
as ab 2 c, is unnecessary. It is, as a rule, enough to point out 
the close analogy with arithmetic, though in a first-year course 
of algebra attention must repeatedly be called to the fact 
that abc is not in form a faithful copy of 345, and that 345 
means 300 + 40 + >. In the main, let early algebraic 
processes grow out of corresponding arithmetical processes. 



Typical Expressions for Factor Resolution 

1. ac + be -f ad + bd = (a + b) (c + d). 

2. x 2 + (a + b)x + ab = (x + a) (x + 6). 

3. acx* + (ad + bc)x + bd = (ax + b) (ex + d). 

These depend on a redistribution of terms, and too much 
care cannot be paid to the teaching of the principle involved. 
We know that 

(a + b)(c + d) = a(c + d) + b(c + d) 
= ac + ad + be + bd, 

and therefore, conversely, 

ac + ad + be + bd = a(c + d) + b(c + d) 

d). 



If then we are given the expression ac + bd + ad + be, 



184 CRAFTSMANSHIP IN MATHEMATICS 

and we rearrange it so that both the a terms come first, we 
have a suitable distribution for finding the factors: 

ac + bd + ad -f- be 
= ac + ad -f- be + bd 
= a(c + d) + b(c + d) 
= (a + b)(c + d). 

Boys are often puzzled about the derivation of the last 
line from the last line but one, but their difficulty is cleared 
up when it is pointed out to them that if they had to multiply 
(a + b) by (c + d), they would begin by writing down 
a(c + d) + b(c + d). 

Emphasis must be laid on this intermediate step of a 
partial redistribution and on how we proceed forwards and 
backwards from it. 

(a -f b -\- c) (d + e) 
= a(d + e) -f b(d + e) f c(d + e) 
= ad -\- ae -j- bd -f- fee + cd + <?, 
which /. = a(rf + r) -f 6(rf + e) + c(rf f- e) 

(a + fe + c) (d + e), with which we began. 

We append two rather harder examples. It is always a 
question of arranging according to the powers of some 
selected letter, though which letter only experience can tell. 

(i) x* + (a + b + c)x + ab + ac 

x 2 -f ax -f bx + # + fl& + # 

Arranging in powers of a, we have 

ax + 0fe + ac + # a + &# + # 
= ^(A; + b + c) + x(x + b + c) 
= (a + x) (x + 6 + c). 

(ii) a 2 -H 206 - 2ac - 36 2 -f 2bc. 

We note the letter c in two terms. Try grouping them 
together. 

Then a 2 + 2ab - 3& 2 - 2ac + 2fo 

= (a 2 + 206 - 36 2 ) - 2c(a - b) 
= (a + 36) (a - b) - 2c(a - b) 
= (a + 36 - 2c) (a - 6). 



ALGEBRAIC MANIPULATION 185 

If boys feel a difficulty about accepting the last line as another 
form of the last line but one, give them an example of the 
reverse kind: 

(a + b) (c + d + e) 

either = (a + b)c + (a + b)d + (a + b)e 

or = (a + b) (c + d) + (a + b)e. 

Both redistributions yield exactly the same result. 
Illustrate with a numerical example: 

47 x 365 

either = (47 x 300) + (47 x 60) + (47 x 5) 
or = (47 X 360) -f (47 X 5), 

i.e. we can perform our multiplication in little bits or in 
bigger bits, just as we please. 

The type x 2 + (a + b)x + ab = (x + a) (x + b) seldom 
gives much trouble. Examples: 

x 2 + Sx + 15. 
x* - 8* + 15. 

The two rules (1) for signs, (2) for determining the coefficients 
of x, should be kept separate. Both admit of very simple 
statement. 

For the first example we begin by writing (x + ) (# + )> 
and for the second example we begin by writing (x ) (x ). 
For both examples we ask the question, What two numbers 
multiplied together give us 15 and when added together 
give us 8? Answer, 5 and 3. Hence the factors (x + 5) (x + 3) 
and (x 5) (x 3). 

Other examples: 

* a + 2x - 15 
x 2 - 2x - 15. 

As the last term is a minus term, the second term of the 
two factors will be of opposite signs. Hence we may begin 
by writing down for each case (x + ) (x ). " Find two 
numbers whose product is 15 and whose difference is 2." 



186 CRAFTSMANSHIP IN MATHEMATICS 

Answer, 5 and 3. " Give the larger number the sign before 
the middle term." Hence we have: 

x* + 2x - 15 = (x + 5) (x - 3) 
x* - 2x - 15 = (x - 5) (x + 3). 

Of course these are mere rules, to be remembered; but they 
should be first worked out from an examination of the different 
products, three or four sets being taken for confirmation 
purposes. 

(x + 3) (x + 5) = x 2 + 8* + 15 

(x - 3) (x - 5) = x 2 - Sx + 15 

(x + 5) (* - 3) = x* + 2* - 15 

(x - 5) (x -f 3) = x* - 2x - 15. 

Help the boys to examine the products and to discover: 

(1) That if the last term of the trinomial is +, the 

signs of both factors are the same, the same as 
the middle term. 

(2) That if the last term of the trinomial is , the 

signs of the two factors are different, the factor 
with the larger number taking the sign of the 
middle term. 

(3) That the last term of the trinomial is always the 

algebraic product of the second terms of the 
two factors (hence the signs). 

(4) That the middle term of the trinomial is always 

the algebraic sum of the second terms of the 
two factors (hence the signs). 

The mere rules must be mastered by all Sets, but experience 
shows that the justification of the rules, by an analysis of a 
series of products, is beyond lower Sets, though upper Sets 
always appreciate them. Do not talk of " proving " the rules. 

The type, acx 2 + (ad + bc)x + bd 

This common type of expression boys generally find 
rather troublesome to factorize. I remember seeing a Fourth 



ALGEBRAIC MANIPULATION 187 

Form trying to factorize 35# 2 59# 48. There had been 
a preliminary discussion on the necessarily long succession 
of " trial " factors, and the 33 boys were actually working 
out with the patience of 33 Jobs the possible combinations, 
the first factors being 35* 1, 7* 1, 5# 1, # 1, 
35* 2, 7x 2, 5* 2, x 2, and so on with 3, 4, 
6, 8, 12, 16, 24, and 48, 80 possible first factors 
in all! Naturally the lesson was not long enough for this 
single set of trials to be completed. In any circumstances 
the particular example would be very difficult for class 
practice. But the " trial " method is unnecessary. All ordinary 
cases can be dealt with by a method which is much simpler. 
Consider the example (6# 2 + 17* + 12) = (3# + 4) 
(2x -f- 3). Let us multiply the factors together in the ordinary 
way. 

3* + 4 

2x H-_3 

ftic a ~~+~8jc 

4- 9* +12 



+ 12 



We might have multiplied out, thus: 

(3* + 4) (2* + 3) 
= 2*(3* + 4) + 3(3* + 4) 
= 6* 2 + 8x -f 9* + 12 
= 6* 2 + 17* -f 12. 

To find the factors, why not reverse this process? 

6* 2 + 17* + 12 

= 6* a + 8* + 9* + 12 
= 2*(3* + 4) + 3(3* + 4) 
= (3* + 4) (2* -f 3). 

Yes, why not? But how could we tell that the ITx in the 
first line should be divided into 8# and 9#, instead of, say, 
into 3# and 14#, or into 5x and 12#? That is the trouble, 
that the only difficulty. How are we to find the two correct 
numbers? 



188 CRAFTSMANSHIP IN MATHEMATICS 

Let us suppose these unknown; call them m and n. 

Now m + n = 17 (that we know). 

And mn = 72. 

[How do we know that? Because 72 is the product of the 
6 and 12 which we obtained (in the multiplication sum) by 
multiplying 3 by 2 and by multiplying 4 by 3; and from these 
same 4 numbers, 3, 2, 4, 3 we obtained the 9 and the 8 also 
in the multiplication sum. Thus the 72 is the product of the 
6 and 12 in the first and third terms of the trinomial.] 

Hence all we have to do is to find two numbers which 
when added together come to 17 and which when multiplied 
together come to 72. The numbers are easily seen to be 
8 and 9, and therefore we now know that the 17* must be 
divided into Sx and 9#. 

Another example: 14# 2 25* + 6. 

Here m + n = 25 and mn = 14 X 6 = 84. By trial, 
the two required numbers are 21 and 4. 

14* 2 - 25* + 6 
= 14* 2 - 21* - 4* + 6 
= 7x(2x - 3) - 2(2* - 3) 
= (7* - 2) (2x - 3). 

Another example: 6# 2 llx 10. 

This time we have to find two numbers whose product 
is 60 and whose sum is 11. The numbers are evidently 
-15 and +4. 

6* 2 - 11* - 10 
= G* 2 - 15* + 4* - 10 
= 3*(2* - 6) + 2(2* - 5) 
= (3* + 2) (2* - 5). 

Thus we have this simple rule. Redistribute the terms of 
the expression , splitting the coefficient of the middle term into 
two parts, m and n, so that m and n is the product of the co- 
efficients in the first and last terms. Then factorize the re- 
distributed product in the usual way. 

For top Sets the rule can be stated more formally from 



ALGEBRAIC MANIPULATION 



189 



the expression acx 2 + (fld + bc)x + bd, where the relations 
stated in the rule are obvious. 

At least top Sets should be made to see how, as regards 
both coefficients and signs, all the different cases may be 
brought under a single rule. Let the general expression be 
ax z + bx + c. Then: divide b into two parts m + n so 
that win = ac. Now let the class apply the rule to all possible 
different cases, say: 

x 2 Sx + 15, 

x 2 2x - 15, 

6:c 2 + 19* + 15, 

6* 2 + x - 15. 

Difficult cases where m and n cannot be obtained readily 
from mn and m + ;/ at once by mental arithmetic may be 
solved quadratically. Examples: 



x* + 2x - 360. 

Write, x 2 + 2x - 360 --= 0. 
/. x 2 + 2x + 1 = 361, 
/. x + 1 == 19, 
.'. x = 18 or 20, 

/. factors = (x 18) (x + 20). 



x 2 + 12* - 405. 

Write, x* + 12x - 405 = 0. 
/. x* -f I2x + 36 = 441, 
.'. x + 6 - +21, 
/. x == -27 or +15, 

/. factors = (x - 15) (x + 27). 



Do not let the pupils look upon these as quadratic equations 
but simply as a plan for finding the factors. Quadratic 
equations will come a little later. The quadratic principle 
may be applied to any case, but more often than not it is 
merely a clumsy substitute for the method first mentioned. 
For instance, consider 6x 2 llx 10. 

6*2 _ 11^ _ 10 = 6(* 2 - Y* - |) 
Solving the quadratic x 2 ~^x J j- = 0, we have, 

#2 __ i i x _|_ /i i\2 _ _|_ /J.i\a 



or f . 



- 11* - 10 = 6* - 



f) 



= (2* - 5) (3* + 2). 



190 CRAFTSMANSHIP IN MATHEMATICS 

Complex derivatives from type forms are a prolific source 
of errors with all but the ablest pupils. Much care is necessary 
in substituting. Example: factorize 8a 3 (a + 2) 3 . 

Type: x* - y* = (x - y) (x* + xy + y 2 ). 
Thus x = 2a; y = a + 2. 

(2a) 3 - (a 4- 2) 3 

= {2 - (a + 2)} {(2a) + 2a(a + 2) + (a + 2)2} 
- (a - 2) (4a 2 + 2a 2 4- 4a + a 2 + 4a + 4) 
= (a - 2) (7a 2 4- 8a 4- 4). 



Product Distribution Generally 

There comes a time, probably towards the end of the 
Upper Fourth year or the beginning of the Lower Fifth, 
when a boy's accumulated facts concerning products must 
be summarized and analysed, and reduced to laws of some 
kind. We will run rapidly over the necessary ground. 

(a 4- b) (c 4- d). Here we have two factors, each of two 
terms. We have to multiply each term of the first factor by 
c and then by d and so we have four terms in all, viz. 

ac 4- be 4- ad 4- bd. 

Compare this with the ordinary arithmetical multiplication. 

37 x 24 

= (30 + 7) (20 4- 4) 

= (30 x 20) 4- (30 x 4) 4- (7 X 20) 4- (7 X 4), 

and show the close analogy. We have and must have four 
products both in the algebra and in the arithmetic. 
For similar reasons: 

(i) (a 4- b 4- c) (d 4- e) will give 6 products. 

(ii) (a 4- b 4- c) (d + e +/) will give 9 products, 
(iii) (a 4- b) (c 4- d) (e +/) will give 8 products, 
(iv) (a + b + c) (d 4- e +/) (g + h + k) will give 27 
products. 



ALGEBRAIC MANIPULATION 191 

The last will be quite clear if it be observed that each of the 
9 products in (ii) has to be multiplied by g, then by h, then 
by k. Clearly, then, if the factors consist of p 9 q, and r terms, 
the number of products will be p X q X r\ and this will be 
quite general. Hence we can tell how many products to expect 
in an algebraic multiplication. 

But in the above cases, all the terms are different. There 
is neither condensation owing to like terms occurring more 
than once, nor reduction owing to terms destroying each 
other. Either or both of these things may happen. 

Consider the product (a -f- b) (a -}- b). By the general 
rule the distribution will give 4 terms' But only 2 different 
letters, a and b, occur in the product, and with these only 3 
really distinct products of 2 factors can be formed with them, 
viz. a 2 , ab y b 2 . Hence, among the 4 terms, at least 1 must 
occur more than once, and, in fact, a X b occurs twice. 
The result of the distribution therefore is a 2 + 2ab + b 2 . 

Thus we may write: 

(a + 6)2 = a 2 + 2ab + b*. 
Similarly (a - 6) 2 = a 2 - 2ab + b 2 . 

In the case of (a + b )(a b), the term ab occurs twice, 
but as the two terms are of opposite signs they destroy each 
other. Nevertheless the main rule still holds good: the 
product really consists of 2 X 2 or 4 terms. 

What are all the possible products of 3 factors that can 
be made with the 2 letters a and 6? Evidently 

aaa, aab, abb, bbb\ 
or, a 3 , a 2 6, a& 2 , 6 3 ; 4 in all. 

Hence in the distribution of (a + 6) 3 , i.e. of (a + b) (a + b) 
(a + b), which by the general rule will give 8 terms, only 
4 really distinct terms can appear. What terms recur and 
how often? 

a 3 and i 3 evidently appear each only once, because to 
get 3 a's or 3 6's we must take one from each bracket, and 
this can be done in only one way. 



192 CRAFTSMANSHIP IN MATHEMATICS 

cPb may be obtained: 

(i) by taking b from the first bracket, and a from each 

of the others; 
(ii) by taking b from the second bracket, and a from 

each of the others; 
(iii) by taking b from the third bracket, and a from each 

of the others. 

ab 2 : the same holds as for a 2 b. 
Thus the result is, 



(a + &) 3 = 3 + 3a 2 6 -f- 3ab* + 6 3 . 

Similarly, (a - fc) 3 - a 5 - 3a*b + 36 2 - b\ 

(a + &) 4 = <z 4 - 



If we remember that the possible binary products of 3 
letters, a, b, c, are a 2 , i 2 , c 2 , ab, ac y be (6 in all), then 

(a + 6 + c) 2 = a 2 + 6 2 + c* + 2ab + 2ac + 2bc. 

The ternary products of 3 letters, a, b, c y are easily enu- 
merated if we first deal with the letter a, writing down 
the terms in which it occurs thrice, then those in which it 
occurs twice, then those in which it occurs once; then deal 
similarly with 6, for such forms as are not already written 
down; then with c. Thus we have (10 in all): 

<2 3 , d*by a 2 c, ab 2 , ac 2 , abc, 

6 3 , b*c, bc\ 

(*. 

Hence, following the rule, we have: 

(a + b + c) 3 - (a + b + c) (a + b + c) (a + b + c) 

-f 3a 2 c + 3ab 2 -f 
Qabc. 



The result may be verified by successive distribution: 

(a + b + c) 3 = (a + 6 + c) 2 (a + b + c) 

= (a 2 + ft 8 + c 2 + 2ab + %ac -f 2bc) (a + b -f- c) 



ALGEBRAIC MANIPULATION 193 

Another example: (b + c) (c + a) (0 + b). 

Here not all the 10 permissible ternary products can occur, 
for a 3 , 6 3 , c 3 are excluded by the nature of the case, a appear- 
ing in only 2 of the brackets, b in only 2, and c in only 2. 

(b + c) (c + a) (a + 6) = fo 2 + b z c + ca 2 + c 2 a -f <z& 2 -f a 2 6 + 2afo. 

But although we do not get the 10 ternary products, we do 
get 8 ( 2 x 2x 2) products, according to the general rule. 
In the product (b c) (c a) (a b) the term abc 
occurs twice but with opposite signs, and there is then a 
further reduction: 

(b - c) (c - a) (a - b) = be 2 - b*c -f ca z c*a + ab* - a*b. 



S Notation 

Upper Sets in the Fifths should be taught to use this 
notation. It is easy to understand, though average boys 
are a little shy of it. " S " stands for, the sum of all the terms 
of the same type as, though its exact meaning in any particular 
case depends on the number of variables that are in question. 

If 2 variables, a and 6, ^Lab means simply ab. 

If 4 variables, #, i, c, d, %ab means ab + ac + ad 

+bc + bd + cd. 

If 2 variables, a, i, 2a 2 6 means a 2 b -f- ab 2 . 
If 3 variables, a, b, c, *Ld l b means a*b + ab 2 + a * c 

+ ac 2 + b 2 c + be 2 . 

The context usually makes clear how many variables are to 
be understood. 

" Choose any one of the terms and place S before it." 
The use of the sign certainly saves labour: thus 



(a + &) 3 = Sa 3 
(a + b + c) 9 = Sa 3 + 3Sa 2 6 + Gabc. 
(b + c) (c 4- a) (a + b) = Z.a*b -f 2o6c. 

(E291) 14 



194 CRAFTSMANSHIP IN MATHEMATICS 

More Complex Forms 

Quick boys soon pick up the method of manipulating 
more complex forms based on those already familiar to 
them, but the slower boys require much practice and should 
not be worried by forms so complicated as to be puzzling. 
The slower boys should always first be given forms with + 
signs only. The added difficulty of negative signs should 
come a little later. 

(i) Type forms, with the addition of coefficients; e.g. 

(a) (3a + 2ft) 3 - (3) 3 + 3(3a) 2 (2b) + 3(30) (2ft) 2 -f (2ft) 3 . 
(P) (a + 2ft + 3c) 2 = &c. 

(ii) Type forms with a mononomial replaced by a 
binomial; e.g. 

Replacing b throughout by (b -f- c) y we have 



(a + b + c) 3 = a 3 + 3a*(b + c) 
= &c. 

(iii) Association of parts of factors of more than 3 
terms; e.g. 

(a) (a + b -f c - d) (a - b + c + <*) 

- {(a + *) + (6 - <0) {(a + c )-(b- d)} 

- (a -f c? - (b- d)\ &c. 

(P) ( + 6 + c) (6 + c - a) (c + a - b) (a + b - c) 

= {(b + c) + a} {(b + c) -a}{a~(b- c)}{a + (b - c)} 

- {(b + c) 2 - a 2 } (a 2 - (6 - c) 2 } 

= (6 2 + 2bc + c 2 - a 2 ) ( 2 - fe 2 -f 2bc - c 2 ) 
= {26c + (6 2 + c 2 - a 2 )} {2fo - (6 2 + c* - a 2 )} 
= (26c) 2 - (6 2 -f c 2 - a 2 ) 2 



- a 4 - fc 4 - c*. 



The type forms are few and are easily remembered, and 
all boys should have them at their finger-ends. The possible 
applications and developments are, of course, very diverse, 



ALGEBRAIC MANIPULATION 



195 



but do not perplex boys with expressions that are beyond 
their manipulative skill at any particular stage. 



Detached Coefficients. First Notions of Manipulation 

Here is a useful general theorem, easy for upper Sets to 
remember. If all the terms of all the factors of a product be 
simple letters unaccompanied by numerical coefficients and 
all +, the sum of the coefficients in the distributed value 
of the product will be / X m X n . . . , where /, m y n, are 
the numbers of the terms of the respective factors. 

Thus in the evaluated products of the following, we have: 





Coefficients. 


Sum of 
Coefficients. 


(a + b)* 


1 + 2+1 


4 = 2 2 


(a + b) 3 


1+3+3+1 


8 = 2 3 


(a + &) 


1 + 4 + 6 + 4 + 1 


16 = 2* 


(a + b) 5 


1 + 5+10+10 + 5+1 


32 = 2 5 


(b + c) (c + a) (a + b) 


1 + 1 + 1 + H- 1 + 1 + 2 


8 = 2 s 



The theorem is useful in connexion with expansions. 

Simple Expansions and First Generalizations therefrom. 
Let an upper Set in a Fourth Form obtain the following 
results by continued multiplication, the second being obtained 
by multiplying the first by (x + #), the third by multiplying 
the second by (x + a) y and so on. 



(x + a) 2 = x 2 + 2xa + a 2 . 
(x + a) 3 = x 3 + 3* 2 a + 3 
(x + a)* = x 4 + 4x 3 a + 
(x + a) 6 - x 5 
(x + a) 6 - x 6 



a 3 . 
+ 4xa 3 + 



15* 2 a 4 + Gxa* + a 6 . 

Now help the boys to generalize, and to establish the usual 
rules. Afterwards, they may work out a few higher expansions 
and see that these follow the same rules. 

1. The power to which we have carried (x + a) gives 



196 CRAFTSMANSHIP IN MATHEMATICS 

the index of the highest terms of the expansion and is there- 
fore the degree of the function. 

2. The function has one term more than that index. 
Thus the expansion (x + #) 4 has 5 terms. 

3. The powers of x run in descending order from the 
first term to the last term but one; the powers of a run in 
ascending order from the second term to the last. (There is 
no objection to writing the first term x n a and the last x Q a n > 
if the class understand that x a = 1. Then both x and a 
appear in every term.) 

4. The dimensions of all the terms are the same and are 
always equal to the power to which (x + a) is carried. 

5. The coefficients follow a regular law. We may detach 
them from their terms (detached coefficients may often be 
usefully considered alone), and place them in order, thus: 



(* + 0) 1 
(* + *) 2 
(x + a) 3 
(* + aY 
(x + a) 5 
(x + a)* 


I 1 
1 2 


1 


1 

6 


1 3 

1 Y~ 
1 5 
1 6 


3 1 
6 4 1 


10 |10 5 
15 20~~~~|15 



The sum of any 2 successive coefficients in any line gives 
the coefficient standing in the next line immediately below 
the second of these. Thus, in the third line the 6 is the sum 
of 3 and 3 in the second line; in the last line, the 15 is the 
sum of 10 and 5 in the fifth line. Show that this is the simple 
result of continued multiplication. For instance, if we multiply 
(x + fl) 3 by (x -f a) we have: 

1 +3+3+1 

1 + 1 

1+3+3+1 

1+3+3+1 
1+4+6+4+1 

The second partial product is arranged one place to the right 
under the first partial product. Thus any coefficient for any 



ALGEBRAIC MANIPULATION 197 

expansion may be found by taking the coefficient of the 
corresponding term in the previous expansion and adding 
to it its predecessor. Let the boys continue the table: they 
like the work. They soon see that when they have written 
the expansion of, say, (x -f- a) 10 , they can immediately write 
down that for (x + 0) 11 ; it is merely a question of carrying 
on the game already begun. 

" There is something still more interesting to learn about 
the coefficients. Consider the expansion of (x + of. The 
coefficient of the second term is 5; we may write it {-. The 
coefficient of the next term is 10, which we may write ^|. 
That of the next term is again 10, which we may write 
1x2x3* ^ nc * so g enera ^y- Examine the other expansions 
and see if a similar rule is followed; for instance, (x + a) Q 

6 * , 6.5 A , 6.5.4 , . 6.5.4.3 9 . 
5 * 4 



/ , N 
(x + a) 6 = 



- ----- __ 

1.4 L.^i.o l.Z.o.4 

6.5.4.3.2 . 6.5.4.3.2.1 6 



With one or two leading questions, the boys will see that the 
coefficient of a 6 is the same as that for X Q , that for xa 5 the 
same as for r r> a, that the coefficient ^ is the same as 
c ~^, and so generally. Let them formulate the obvious rule 
for themselves. 

Let them write down the first few terms of such an 
expansion as (x + fl) 20 - 

First they write the terms without coefficients: 

x 20 + x l9 a + x l *a* + &c. 

Then they work out their coefficients and insert them: 
. ,20 . 20.19 , Q 



The object of all this is not to teach the Binomial Theorem: 
that will come later. It is to impress boys with the wonderful 
simplicity and regularity that underlies algebraic mani- 



198 CRAFTSMANSHIP IN MATHEMATICS 

pulation. Never mind the generalized form (x + a) n , yet. 
Never mind the general term. Never mind nCr. When 
these things are taken up later, the way will have been paved, 
and the work will give little trouble. 



The Remainder Theorem 

This theorem must be known in order that the Factor 
Theorem may be clearly understood. It may be approached in 
this way. We know that (x 5) is a factor of (x 2 -\-x 30), 
and in order to find out the other factor we may conveniently 
set out the process of division, exactly as in arithmetic. 

x - 5)* 2 + x 30(# + 6 
x 2 - 5* 

0* 30 
6* - 30 



Of course there is no " remainder " (R), but if (x 5) was 
not a factor there would be a R. Divide (3# 2 2x + 4) 
by (x - 5). 

x - 5)3* 2 - 2* -f 4(3* f 13 

??L_Z. I5x 

13*+ 4 

13* - 65 

69 = R. 

The remainder is 69, and by analogy with arithmetic we know 
that 

Dividend = (Quotient X Divisor) -f R. 

Suppose we had to divide (x 2 + px + q) by (x a). 

x a)x* px + q(x + (a p) 
x 2 ax 

x(a p) + q 

x(a - p) - a(a - p) 

a(a p) -f q = R. 

Note that a(a p) + q really is the remainder, for it does 



ALGEBRAIC MANIPULATION 199 

not involve x, and we cannot proceed with the division any 
farther. 

Now let us set out in this way the previous example, 
treating the figures as if they were letters, and not actually 
multiplying and subtracting as we did before. 

x - 5)3x 2 - 2x + 4(3* + (3.5 - 2) 



x(3.5 - 2) + 4 

x(3.5 - 2) - 5(3.5 - 2) 

5(3.5 - 2) + 4 = R. 

As might be expected, the R is the same, viz. 69. 

Now compare the Remainders and the Dividends in both 
the last examples. 

Dividend = x 2 px + q. I Dividend = 3# a 2x + 4. 
Remainder = a 2 pa + q. \ Remainder = 3.5 2 2.5 + 4. 

Clearly, then, if the remainder was the only thing we wanted, 
we could have written it down at once, for it is exactly of the 
same form as the dividend. We merely have to substitute 
for the x in the dividend the second term of the divisor 
(a in the first case, 5 in the second), treating these, however, 
as if they were positive. 

Give the pupils several examples, and convince them of 
the truth of the rule. 

The Remainder " Theorem ", as it is called, provides us 
with a simple means of calculating the remainder of a particular 
kind of division sum in algebra, without actually performing 
the division. 

The particular kind of division sum is that in which the 
divisor and the dividend are functions of the same letter 
(say #), and in which the divisor is a linear expression such 
as (x 5) with unity as the coefficient of x. 

Example: if we divide 

(* 3 - 7* 2 + 5* - 1) by (x - 9), the R is 
(9 3 - 7.9 2 + 5.9 - 1) = 206. 

We have merely substituted 9 for x in the dividend. 



200 CRAFTSMANSHIP IN MATHEMATICS 

Hence: 

(1) The Theorem. When a function of x is divided 
by (x a), the R is obtained by substituting 
a for x in the function. 

(2) The why of it. We know that, 

Dividend = (Quotient X Divisor) -f R. 

If we make a equal to x, the divisor (x a) = 0. 
/. Dividend = (Quotient X 0) -f R, 

-R, 
i.e. by substituting a for x in the Dividend, we have the R. 

The Factor Theorem 

What is the remainder when we divide (# 2 7x -f- 10) 
by (x - 5)? 

Substituting 5 for x in x 2 Ix -|- 10, 

we have 5 2 7.5 -\- 10 

- 0. 

Since R = 0, (x 5) divides exactly into (x 2 7x + 10) 
and is therefore a factor of it. 

Thus we have a method of finding out whether an ex- 
pression of the type (x a) is a factor of a given expression 

Example: Is (3* 3 - 2* 2 - 7* - 2) divisible by (x - 2) (x + 1)? 

Writing 2 for x in the first expression, we have 

24 814 - 2 -= 0. Hence (x 2) is a factor, 

Again, writing 1 for x we have 

-3-2 + 7-2-0. Hence (x + 1) is a factor. 

Homogeneous Expressions 

A homogeneous expression is one in which all the terms 
have the same dimensions; e.g. 

x* + xy + y*> or a 3 + 6 3 + c* + Sabc. 



ALGEBRAIC MANIPULATION 201 

It may sometimes be necessary to state what letters are re- 
garded as giving dimensions; e.g. x 3 + ax 2 y + 2xy 2 + 3y 3 
is homogeneous in x and y but is not homogeneous if a is 
regarded as having dimensions. 

Obviously the product of two homogeneous expressions 
is itself homogeneous. 

The only homogeneous integral functions of x and y 
of the first and second degrees are, 

Ax 4- By, 

Ax 2 4- "Rxy + Qy 2 . 

For 3 variables the corresponding functions are, 

Ax -|- I*y -1- C#, 

Ax 2 4- By 2 4- C* 2 4- D;vx? + Esx + Fxy. 

The class may usefully write down functions of the third 
degree. Upper Form boys should be thoroughly familiar 
with all such general expressions. 



Symmetry 

A function which is unaltered by the interchange of any 
two of its variables is said to be symmetrical with regard 
to these variables', e.g. x 2 xy 4- y 2 is symmetrical with 
regard to x and y] (y + #) (z + x) (x -\- y) is symmetrical 
with regard to x, y, and z. But x 2 y + y 2 z 4- ~ 2 # is not a 
symmetrical function of x, y, and #, for the 3 interchanges 
x with y, y with z, z with x, give, respectively, 

y z x + ff 2 * + # 2 }>, 
x 2 z + # 2 ;y 4- :V 2 # 
z*y 4- y 2 * 4- A, 

and although all these are equal to each other, none of them 
is equal to the original expression. 

But the term " symmetry " is not used in quite the 
same sense by all writers in algebra. " Cyclic symmetry " 
expresses a much clearer connotation. 




202 CRAFTSMANSHIP IN MATHEMATICS 

Cyclic Expressions 

An expression is said to be " cyclic " with regard to the 
letters a y b, c, d, . . . k> arranged in this order, when it is 
unaltered by changing a into b, b into c, . . . k into a. Thus 
the expression a 2 b + b 2 c + c 2 d + d 2 a is cyclic with regard 
to the letters a, i, , d, arranged in this order, for by inter- 
changing a into b, b into c, c into d, d into a, we get 
b 2 c -f c 2 d + d 2 a + a 2 b, the same as be- 
fore. Note that the first term is changed 
to the second, the second to the third, 
and so on. It is merely a question of 
beginning at a different point on the 
circle, but always going round in the 
same direction. 

Fig. 60 If in a cyclic expression a term of 

some particular type occurs, the terms 
which can be derived from this by cyclic interchange must 
also occur, and the coefficients of these terms must be equal. 
Thus, if x y y, and % are the variables, and the term x 2 y 
occurs, then all the terms x 2 y y x 2 z y y 2 z, y 2 x y % 2 x, z*y must 
occur. The cycle is easily seen if the six terms are thus 
collected up: 

x*(y + *) + y*(* + x) + z\x + y). 

Expressions which are unaltered by a cyclical change of the 
letters involved in them are called cyclically symmetrical. 
Thus (b c) (c a) (a b) is cyclically symmetrical, since 
it is unaltered by changing a into i, b into c, and c into a y 
that is " by starting at a different point in the circle ". 

Legitimate Arguments from Cyclical Symmetry 

Find the factors of 

a\b - c) + b*(c -a) + c\a - b). . . . (i) 

Here is the solution from one of the best textbooks we have. 



ALGEBRAIC MANIPULATION 203 

" If we put b = c in the expression, the result is zero, 
and it therefore follows from the Remainder theorem that 
(b c) is a factor. 

" In a similar manner we can prove that (c a) and 
(a b) are factors. 

" Now the given expression is of the third degree; it 
can therefore have only 3 factors. 

" Hence the expression is equal to 

N(6 - c) (c - a) (a - 6), (ii) 

where N is some number which is always the same for all 
values of a, ft, c. 

" We can find N by giving values to a, ft, c, Thus, let 
a = 0, ft = 1, <: = 2; then (i) = -2, and (ii) = +2N. 
Hence N = -I. 

" Therefore the factors are (b c) (c a) (a ft)." 

This argument is open to criticism. It is wholly un- 
necessary to say, " in a similar manner we can prove that 
(c a) and (a ft) are factors". Once we know that (ft c) 
is a factor, it follows at once that (c a) and (a ft) are 
factors. What applies to (ft c) must apply to (c a) and 
(a ft). This is the very essence of cyclical symmetry. 
Nay, it is the very essence of all algebraic manipulation. 
That (c a) and (a ft) are also factors requires no argu- 
ment of any sort or kind, except, " it follows from cyclical 
symmetry "; and no further argument should be tolerated. 

Another example. Find the factors of a 3 (b c) + b\c a) 
+ c 3 (a ft). As in the last example (ft c), (c #), (a ft) 
are all factors. Now the given expression is of the fourth 
dimension; hence there must be a fourth factor, and that of 
the first dimension. Since this factor must be symmetrical 
with respect to #, ft, , it is necessarily (a + ft + c). Thus 
the required factors are 

N(6 -c)(c-a)(a-b)(a + b + c), 

N being found in the usual way. Any sort of more elaborate 
process or argument should be sharply criticized. 



204 CRAFTSMANSHIP IN MATHEMATICS 

Another example. Find the product of (a + b + c) 

(a* _|_ #2 _j_ c z _ fa _ ca __ a b^ 

Each of the two factors is symmetrical in a, 6, c, and 
therefore the product will be symmetrical in a, 6, . 

Obviously the term a 3 occurs with the coefficient unity; 
hence the same must be true of 6 3 and c 3 . 

Obviously, too, the term Ire has the coefficient 0; hence 
by symmetry the five other terms Ire, c 2 a y ca 2 , ab 2 y a 2 b belong- 
ing to the same group must have the coefficient 0. 

Lastly, the term abc is obtained (i) by taking a from 
the first bracket and be from the second; hence it is also 
obtained (ii) by taking 6, and (iii) by taking c, from the first 
bracket. Thus the term abc must have the coefficient 3. 
Hence the product 

= a 3 + fe 3 + c 3 - 3abc. 

Boys should gain complete confidence in arguments from 
symmetry. In at least the A Sets of the Fifth Form, cumbrous 
processes should be prohibited whenever arguments from 
symmetry are possible. 



Identities to be Learnt 

The following identities should be at the finger ends of 
all Fifth Form boys. 

1. (b-c) + (c-a) + (a-b) = 0. 

2. a(b - c) -f b(c - a) -f c(a - b) = 0. 

3. (a + b -|- c) 2 = a 2 f b 2 + c 2 + 2a6 + 2bc -f 2ca. 

4. (a + b -f c) 3 - a 3 4- 6 3 -f c 3 + 3b 2 c + 3bc 2 + 3c 2 a + 3oi a 

-f 3a 2 Z> f- M 2 + Gabc. 

5. (a + b + c) (a 2 + b 2 + c 2 - be ca ab) = a 3 -f ft 3 -f- & 

-Zabc. 

6. (6 - c) (c - a) (a - b) = - 2 (6 - r) - 2 (c - a) - c 2 (a b) 

= fo(6 c) ra(c a) ab(a b). 
1. (b + c) (c + a) ( + 6) - a(6 + c) + b 2 (c + a) + c 2 (a + 6) 



ALGEBRAIC MANIPULATION 205 

and perhaps, 

8. (a + b + c) (a 2 + 6 2 + c 2 ) 

= bc(b + c) 4 ca(c 4 a) + ab(a + b) + a 3 + b 3 4 c 3 . 

9. (a + b 4- c) (be 4 ca 4 aft) 

= 2 (6 4 c) 4 ft 2 (c + a) 4- 2 (a + ft) 4 3fo. 
10. (a 4 ft 4 c) (ft 4 c - a) (c 4 a - ft) (a 4 6 - c) 
= 2ft 2 c 2 4 2c 2 a 2 4 2a 2 ft 2 - a 4 - ft 4 - c 4 . 

Books to consult: 

1. Textbook of Algebra (2 vols.), Chrystal (still the leading work 

in the subject). 

2. A New Algebra, Barnard and Child. 



CHAPTER XIX 

Algebraic Equations 

Equations of Different Degrees 

" Either . . . or?" 

More than once I have heard a small boy round on his 
teacher for this kind of argument: 

Solve the equation x 2 7x + 12 = 0. " Factorizing, we 
have (x 4) (x 3) = 0. Hence either (x 4) or (x 3) 
must be zero, i.e. x must be either 4 or 3, and therefore 
both 4 and 3 must be roots of the equation." 

Says the boy: " You said either (x 4) or (x 3) must 
be zero; how then can it follow that x is both 4 and 3?" 

The criticism is just, for the reasoning is faulty. 

A formal approach to equations may be successfully 
made by such general arguments as follows. 

It is advisable in the first place to distinguish between 
an equation and an identity, and consistently to use the same 
form of words when referring to them. For instance: "When 
two expressions are equal for all the values of the quantities 



206 CRAFTSMANSHIP IN MATHEMATICS 

involved, the statement of their equality is called an identity" 
e.g. that m (n p) = m n -{- p is true for all values of 
the letters m, n y and p. 

" But when two expressions are equal for only particular 
values of the quantities involved, the statement of their 
equality is called an equation." Thus # + 7 = 10 is an 
equation; it is true only where x 3. 

If in an equation we bring all the terms from the right- 
hand side to the left-hand side, and equate the whole to 0, 

e -8- x + 7 - 10 = 0, 

then by giving x its own particular value, the expression 
" vanishes ", e.g. 

3 + 7 _ 10 = 0, 

i.e. 3 + 7 10 is seen really to be 0, and has therefore 
" vanished ". 

The value of the unknown quantity that makes the two 
sides of an equation equal is said to satisfy the equation. 
The process of finding that value, the root, is called solving 
the equation. 

Consider the equation 3(# 2) = 2(x 1). 

For what value of x is 3(x 2) equal to (2x 1)? 

Try a few values, say the numbers 1 to 10. The only one 
of the ten that makes the expressions equal is 4, i.e. 

3(4 - 2) = 2(4 - 1), 

and so we say that 4 is the root of the equation. 
If we simplify the original equation, we have 

3* - 6 = 2x - 2, 
/. 3* - 2x = 6 2, 

.". x 4, as expected. 

We may write #=4 as x 4 =^ 0, and when in the ex- 
pression x 4 we write 4 for x, the expression vanishes, 
for 4 - 4 = 0. 

Again, for what value does x 2 - x = 6? 

Try a few numbers as before. We find that in this case 



ALGEBRAIC EQUATIONS 207 

there are two values which satisfy the equation, viz. 3 and 
2, and that there are no others. Substituting, we have 

3 2 - 3 - 6, 
and (-2) 2 - (-2) = 6. 

If we write the equation in the form x 2 x 6 = 0, the 
expression on the left-hand side vanishes when we write 
either x = 3 or x = 2. Thus 

3 2 - 3 - 6 = 0, 
and (-2) 2 - (-2) - 6 - 0, 

and it does not vanish for any other value. 

With equations of the second degree, we may always 
find two values of x that will satisfy the equation. 

Since x 2 x 6 = 0, 

and since x* x 6 = (x 3) (x -f- 2), 

.'. (x - 3) (* + 2) = 0. 

Now a product cannot be equal to zero unless one of the 
factors is equal to zero, and hence (x 3) (x +2) can be 
equal to zero only (1) when x 3 0, and (2) when x + 2 
= 0, and never otherwise. Thus when we have (x 3) 
(x + 2) = 0, we may equate the two factors equal to 
separately, solve the two simple equations, and obtain the 
two roots: 

Since x 3 = 0, .*. x = 3, 
and since x -f 2 = 0, .*. x = 2; as before. 

Thus we have a method of solving a quadratic equation. 
Bring all the terms to the left-hand side and equate to 0; 
break up the expression into factors, and equate each of these 
to 0; solve the two resulting simple equations. 

Note that a quadratic equation has two roots. 

Here is a quotation from a well-known textbook: " If 
the product of two quantities is nothing, one of the quantities 
is nothing." One objection to this phraseology is that " noth- 
ing " is not a mathematical term. 

Suppose we have an equation of the third degree, a 



208 CRAFTSMANSHIP IN MATHEMATICS 

cubic equation as it is called, say, x* 6# 2 llx = 6. 
Bring all the terms to the left-hand side and equate to 0. 
By a series of trials we may discover that there are 3 and 
only 3 values of x which will make the expression on the left- 
hand side vanish, viz. 1, 2, and 3 Thus 

# 3 - 6* 2 I- 11* --0 = 
I 3 6(l) 2 + 11 6 = 
23 _ 6(2) 2 + 22-6-0 
33 _ 6 (3)2 4-33-6-0. 



Hence the roots of the equation are 1 , 2, 3. But the trials 
would have been tedious. Let us factorize as before. 

Since x* - 6* 2 + 11* - 6 == 0, 
(x -!)(*- 2) (* - 3) = 0. 

A product of factors can be equal to only when one 
of its terms is equal to 0. Obviously, in the equation, this 
may happen in three different ways, when (x 1) = 0, when 
(x 2) = 0, when (x 3) -- 0, and in no other way. If 
then we solve these three simple equations, we get x = 1 
or 2 or 3, as before. A cubic equation has three roots. 

Suppose we have an equation of the fourth degree, a 
" biquadratic " equation as it is called. It is quite easy to 
solve if we can factorize the expression made by bringing 
all the terms to the left-hand side. Usually this is a difficult 
job, but here is an easy one. 

x* + Qx* -f 38* = 8jc 3 + 40, 
/. #* - 8* 3 + 9* a + 38* - 40 - 0, 
/. (x -!)(* + 2) (x - 4) (x - 5) - 0. 

This product can be zero only if one of its factors is zero. 
This can happen in 4 ways, and only in 4, viz. when x 1 == 0, 
#-j-2 = 0,# 4 = 0, x 5 = 0. Thus the orginal equation 
is equal to these 4 separate simple equations, and the roots are 
1, -2, 4, 5. 

Thus a biquadratic equation has 4 roots. And so we 
might go on. 

The general rule for solving an equation of a degree 



ALGEBRAIC EQUATIONS 209 

beyond the first is, then, to bring all the terms to the left- 
hand side, to reduce the resulting expression to a series of 
linear factors, to equate each of these to 0, and then to solve 
them as simple equations. 

It is therefore clear that the roots of an equation of any 
degree may be written down at once, provided we can resolve 
into linear factors the expression which results from bringing 
all the terms of the equation to the left-hand side. Generally 
speaking, the trouble is to find the factors, and it is often 
necessary to resort to indirect methods. 

The Need for Verifying Roots 

When solving equations, we frequently adopt the device 
of multiplying or dividing both sides by some quantity, 
and sometimes we square, or take the square root of, each 
of the two sides. Is this always allowable? 

Consider an equation of the simplest form, one having only 
one solution, say x 3 = 2. Since x 3 2, x 5 =0, 
and .-. x = 5. Let us multiply both sides of the original 
equation by, say, (x 6). Thus 

(x - 3) ( X - 6) = 2(* - 6), 
/. x 2 9x -f 18 = 2.v 12, 
/. x 2 - llx H- 30 - 0, 
/. (x - 5) (x - 6) = 0, 
.". the values of x are 5 and 6. 

Thus by introducing the factor (x 6) we have trans- 
formed the equation into another completely different. The 
new root 6 does not satisfy the original equation. Evidently 
when we have solved an equation we must see if the roots 
really satisfy the equation. 

Another example: consider the very simple equation 
x = 3. Square both sides, 

* 2 = 9, 

/. *2 _ 9 = 0, 
/. (x + 3) (x - 3) = 0. 

Hence there are 2 roots, +3 and 3, as compared with 

(B291) 15 



210 CRAFTSMANSHIP IN MATHEMATICS 

only one root (+3) in the original equation. Thus the 
squaring has introduced an extraneous root. 
Another example: 

3* V* 2 ^~2~4 = 16. 

/. 3* - 16 = V^-24 
:. 9* a - 96* + 256 = x 2 - 24 
/. 8* 2 - 96* -f 280 = 0, 
.'. x 2 - I2x + 35 = 0, 
... (x - 7) (* - 6) = 0, 

/. x = 7 or 6. 

But on examination we find that only 7 satisfies the original 
equation; 5 does not. Hence there is only one root. We 
seem to have solved the equation in the usuial way: have we 
done anything wrong? Let us see if by working our way back- 
wards we can discover any sort of mistake. 

(x - 5) (x - 7) = 0, 

/. x 2 I2x + 35 = 0. 

Multiplying by 8, 8* 2 96* -f 280 = 0. 

Adding x 2 24 to each side, 9* 2 96* + 256 * 2 - 24. 

Extracting the square root of each side, 3* 16 = + v/ * 2 24. 
.'. 3* T v ^-l4 = 16. 

The steps are exactly the same until we come to the last but 
one; then we had to prefix the double sign. Hence at the 
second step in our forward process, we really introduced a 
new and extraneous root, since the square of + V# 2 24 
is also the square of V# 2 24! Thus from that step on- 
wards, the equations ceased to represent the original equation. 
When we multiply or divide by ordinary arithmetical 
numbers, no difficulty will arise. When we multiply or 
divide by an algebraic expression, we sometimes run a risk. 
What is wrong with the following, for instance? 

Suppose * = y. 

Then x 2 = ocy, 

.'. x 2 y 2 xy y* 9 
.' (x + y) (x - y) = *(* - y), 
:. x + y = *, 
/. x -f x = x, 

:. 2=1, which is absurd. 



ALGEBRAIC EQUATIONS 211 

We have divided both sides in the fourth line by (x y), 
i.e. by (x x), i.e. 0. This is quite illegitimate, and it in- 
evitably leads to an absurdity. Can you see now why our 
first example went wrong? We had, really, (x 5) = 0; 
and then (x 5) (x - 6) = Q(x 6), though we did not 
show it this way. 
Another example: 

2 x , 1 A 

-f 2 4- ----- =0. 



- 
x 2 1 x 1 

Multiply by * 2 - 1, the L.C.M., 

x* ~ 3x -f 2(x* - 1) + * -h 1 = 0, 

/. 3* 8 - 2x - 1 - 0, 

/. (3* 4- 1) (x - 1) - 0, 

.*. x = J and 1. 

But by testing we find that 1 is not a value of the original 
equation and must therefore be rejected. Multiplying by 
(x 2 i) led to this trouble. Here is a more correct way of 
solving: 



- 3* 



-2 = 0, 
-^ + 2 = 0, 

j o rv 

.". x = 5, the only root. 



The former method is quite acceptable, provided the roots 
found are checked, and that one is rejected if found un- 
acceptable. 

Strictly speaking, either ... or are " disjunctive " 
they therefore suggest alternatives. But sometimes they 



212 CRAFTSMANSHIP IN MATHEMATICS 

are equivalent to both . . . and] or, alike . . . and\ and it 
is this exceptional use which more correctly represents the 
algebraic argument. But the use of either ... or in con- 
nexion with equations is best avoided. When a boy says 
" either 6 or 5 " he naturally thinks that if one is accepted 
the other is necessarily rejected. 



The Theory of Quadratics 

The work on equations should be closely associated with 
the work on graphs. The graph helps to elucidate all sorts 
of difficulties. See, for instance, fig. 46, p. 161, in connexion 
with the " either ... or " argument. 

The elementary theory of quadratics, as far as it is 
necessary for a Fifth Form, seldom gives trouble. The more 
elementary facts should be known thoroughly and should 
be consistently used for checking and other purposes. But 
do not forget that the quadratic function is of far greater 
importance than the quadratic equation (see the chapter on 

Gra p s) - 



. - , , 

The formula - == --- may be used as other 
2a 

formulae are used, but it should not be used as the stock 
method of solving quadratics; boys are apt to forget its 
significance if used in that way. They should clearly realize 
that the formula represents the roots of the equation ax 2 + bx 
+ c 0, and that these roots are real and different, real 
and equal, or unreal and different, according as the dis- 
criminant b 2 4.ac is +, 0, or . 

The pupils should frequently make use of the further 
facts that if x l and x 2 arc the roots of the equation ax 2 + bx 

+ c = 0, then x 1 + x 2 = , and x^ = -. The method 



a a 



r i - u r * Vb 2 4ac -j , 
or evaluating these from - - should be re- 

membered. a 



ALGEBRAIC EQUATIONS 213 

Equations Solved like Quadratics 

Group the common types together. They are usually 
easy, though attention must be paid to all the roots involved; 
the boys are apt to overlook some of them. We append 
examples of the main type, and add the sort of hint that 
ought to suffice to enable the boys to set to work. 

1. x l 



Write * 2 = y, and solve for y\ then from the 2 values of y 
obtain the 4 for x. 

2. (x 2 + 2) 2 - 29(* 2 -J- 2) | 198 - 0. 

Write # 2 + 2 = y and solve for y. 

3. 2x z - 4x -f 3 Vx 2 ~2x -\ 6 - 15 = 0. 

/. 2(x* - 2* -f 6) + 3V* 2 ^-"2* +7> - 27 = 0. 
Write Vx 2 2x 4- 6 = y and solve for y. 

4. (x -l)(x- 3) (x - 5) (x - 7) = 9, 

.'. (x !)(* 7) (x - 3) (x - 5) - 9 - 0, 
/. (x* - 8* + 7) (^c 2 - 8^ + 15) - 9 = 0. 
Write x 2 8x = y, and solve for y. 

x -f 4 A: 4 9 + ^: 9 A: 



5. 



- 4 i + 4 9 "i 9 + 
16* 36* 



16 81 x 2 
4 9 



6. X*+ - - 11 -0, 

/ **2 



(obviously * is a root), 
(obviously 2 more roots). 



/. ** - II* 2 -f 18 = 0. 

7. x z + 1 = 0. (Factorize.) 

8. x 6 7x* 8 =r 0. (Factorize.) 

9. 7x* - I3x z + Zx + 3 = 0. (Factorize: (* - 1) evidently 

a factor.) 

Boys soon see through all these types and solve examples 
fairly readily. 



214 CRAFTSMANSHIP IN MATHEMATICS 

Simultaneous Equations 

Do not spend much time over these, unless you are un- 
lucky enough to have to prepare for an unintelligent exam- 
ination in which far-fetched examples are given. Let a 
result of each main type be graphed (see the Chapter on 
Graphs) and all the roots be pictorially explained. 

Teach the pupils to pair off the roots correctly. 

We append an example of the commoner types. 

1. x -\ y = 1 

XV 14 Show that once we know the value of x -\- y 

and x y, we may obtain the separate 

2. x 2 -f y 2 - 1^ values of x and y by mere addition and 

xy 6 subtraction; and that x -\- y can always be 

obtained from x 2 -f 2xy ~|- y 2 , and x y 

3. X - y = 2 from x * _ 2xy + yZf 



4. x 3 -! y 1 = 152 

X \- y - S By division we obtain quotients which 

enable us to proceed as in examples 

5. *3-j,3 = 98 Ito3 . 

x 2 -f xy f y 2 = 49 

Ity division, we obtain x 2 ~ xy + y a , 

6. ^ 4 + ^j; 2 -\-y* -- 133 which, subtracted from x 2 -f ^ 

Jt: 2 4- ^ry + jy 2 ^ 19 -I- y 2 , gives us ;ry. Then as in 

examples 1 to 5. 

7. 3x* + 4xy + 5); 2 ^=31 From the second express y in terms 

X ~\- 2y 5 of x, and substitute in the first. 

Expressions are homogeneous. Con- 
vert into fractions, simplify, and 

8. X 2 + 3xy y 2 = 9 factorize. Thus we have: 
2x 2 - 2xy -f 3y 2 = 7 ll* a - 39^ + 34>' 2 = 0, 

/. (11*- lly) (x - 2y) = 0, 
.'. x = ^{y and 2y\ &c. 



All these types are easily taught and remembered. It is 
waste of time for boys to be given the far-fetched and 
exceptional types worked out (often elegantly it is true) 
in the textbooks. School life is not long enough. 



ALGEBRAIC EQUATIONS 215 

Problems producing Equations 

These have been given a place greatly beyond their value, 
and important mathematical principles are often treated 
rather superficially in order that more time may be devoted 
to " problems ". Unfortunately, however, problems have 
become entrenched in all mathematical examinations, and 
there is nothing for it but to teach boys how to solve them. 
And, after all, problems do test boys' knowledge of certain 
principles, and a correct solution is always a source of satis- 
faction. 

The veriest tyro of a teacher can write out on the black- 
board the solution of a problem which the boys themselves 
have been unable to solve. But what do the boys gain from 
that? The mere setting out of a solution deductively, after 
the manner of a proposition in Euclid, gives the boys no 
inner light at all. The boys want to be initiated into a plan 
of effective attack, to be taught how to analyse and how to 
utilize the data of a problem, to be told exactly how the 
teacher himself discovered the solution. 

Be it remembered that a solution most suitable for a 
class of boys is by no means necessarily the " neat " solution 
so dear to the heart of a mathematician. 

The main difficulty felt by boys in solving most algebra 
problems is the translation of the words of the problem 
into suitable equating formulae. Much practice is necessary 
if facility in this translation is to be gained. Once expressed 
in algebraic form, the equation is generally easy of solution, 

The boy knows that an equation consists of two parts 
connected together by the sign =. The first thing to search 
for in a given problem is therefore the word " equal ", or 
some words which imply " equal ", or such words as " greater 
than " or " less than ". If the problem concerns money 
matters, the boy may be able to dig out of his own know- 
ledge some relation of equality, e.g. 

Cost price + profit = selling price; 



216 CRAFTSMANSHIP IN MATHEMATICS 

or, if he is dealing with racing problems, he may be able to 
utilize the already familiar relation: 

distance = speed X time; 

or, if he is dealing with a clock sum, he may be able to split 
up a component angle in two different ways, and so equate 

a + P y ~f 8 (or some modification of this). 
We append a few problems, with teaching hints. 

1. If 4 be added to a certain number, and the sum be 
multiplied by 5, the product will be equal to the number added 
to 32. Find the number. 

The question tells us, 

a product the number + 32 (i) 

Let us try to arrange our equation accordingly. 

What have we to find? A number. Then let x represent 
the number. The " product " is 5 times the sum of x and 4; 
how shall we write this down? 5(# + 4). (i) tells us that 
this product is equal to x + 32; 

/. 5(# + 4) = x + 32. /. x = 3. 

2. Find a number such that if it be multiplied by 5, and 2 
be taken from the product, one-half the remainder shall exceed 
the number by 5. 

The question says 

half a remainder exceeds the number by 5, 
i.e. half a remainder = the number -\- 5 (i) 

How shall we represent the number? by x. 5 times the 
number? 5x. What is the remainder when 2 is taken from 
this product? 6x 2. What is half this remainder? %(5x 2). 
Then how from (i) can we make up our equation? 

- 2) = x + 5. .'. x = 4. 



ALGEBRAIC EQUATIONS 217 

3. A man spent 10 of his money, and afterwards one- 
quarter of the remainder. He had 30 left. How much had 
he at first? 

The word left suggests the -relation: 

(money at first) (expenditure) = 30. . . . (i) 

Let us try to arrange our equation accordingly. 

Let x represent the number of pounds he had at first. Then 

x expenditure 30 (ii) 

What is the expenditure? 

First expenditure = 10; /. x 10 = remainder. 
Second expenditure = } of remainder = -J(*v 10), 
.'. Total expenditure = 10 -f \(x 10). 

Now we may substitute this in (ii): 

.'. x - {10 + \(x - 10)} - 30. /. X = 50. 

4. A man buys a flock of sheep at 3 a head, and turns 
them into a field to graze for 3 months, for which he is charged 
45s. a score. He then sells them at 3, 10s. a head, and so 
makes a clear profit of 77, 10s. How many sheep were there 
in the flock? 

Clearly: 

(Money laid out) f (profit) (Proceeds of sale). . (i) 

What have we to find? The number of sheep bought. 

x 
Let x represent no. of sheep bought; then = no. of 

scores of sheep bought. 

1. Money laid out: 

(a) Cost of x sheep at 3 each = 3x pounds 

(b) Cost of grazing -- scores of sheep at 2 \ a score = f X 2 

/. total money laid out = 3# -f- ( ~ X 2H . . . (ii) 

2. Proceeds of sale: 

Sale of x sheep at 3 J each = 3J# pounds. , . . (iii) 



218 CRAFTSMANSHIP IN MATHEMATICS 

According to (i), (ii) + ?7| = (iii), 

i.e. 3* + ~ X 2i + 77J - 3*. /. X - 200 



5. A boy was born in March. On the 18th of April he 
was 5 times as many days old as the month of March was on 
the day before his birth. Find his date of birth.* 

This examination absurdity is simple enough, once the 
wording is unravelled. Note that if a boy is born, say, on 
4th May, he is 20 days old on 24th of May. It is a case of 
simple subtraction. 

In the problem we have to deal with two ages, expressed 
in days: 

(i) the age of the boy on J8th of April, 

(ii) the age of March on the day before the boy was born 

The former = 5 times the latter. Hence we can 

make up our equation ........ (i) 

(i) The age of the boy on 18th of April: 

Let the boy be born on the #th of March. 

By the end of March he is (31 x) days old. 

By April 18th he is (31 * + 18) days old ..... (ii) 

(ii) The age of March on the day before the boy was born. 

The boy was born on the #th day of March. 

Hence March was then x days old. 

The day before that, March was (x 1) days old. . . (iii) 

From (i) we know that (ii) is 5 times (iii). 

i.e. (31 - x) + 18 = 5(* - 1). /. x = 9. 

6. The 3 hands of a watch are all pivoted together centrally. 
When first after 12.0 will the seconds hand, produced back- 
wards, bisect the angle between the other 2 hands? 

We have to remember that the seconds hand moves 60 

The problem is not well worded. For instance, March is not, strictly, nine 
days old until midnight on March Qth. We have assumed that the boy was born 
at midnight, and we have reckoned ages from midnight. 



ALGEBRAIC EQUATIONS 



219 



times as fast as the minute hand, and the minute hand 12 
times as fast as the hour hand. Thus the relative speeds are 
720 : 12 : 1. 

At noon (N) all the N 

hands are together. The 
watch circumference is 
divided into 60 equal arcs, 
and we may measure the 
angles in terms of these 
arcs. Let the seconds 
hand move round to its 
position S* in x seconds; 
i.e. arc NS = x. Since 
the minute hand also 
moves round to its posi- 
tion M in x seconds, the 
x 
60' 

And since the hour hand also moves round to its position 

v 

H, in x seconds, the arc NH measures - . 

Now the seconds hand OS produced backwards, making 
OS', bisects the angle HOM; i.e. the arc HS' = the arc S'M. 

We ought therefore to be able to make up an equation 
by means of the pieces of arc between N and M, e.g. 

NM = NS' 4- S'M (i) 

We know that NM = %- , 




arc NM measures 



Fig. 61 



- NH) = 



that NS' = x - 30, 
and that S'M = JHM - 

.'. from (i) we have ~ = (x 30) + |f ~ -~\, 

:. x = 30^^ (sees, after 12.0). 
The angles in the figure are necessarily much exaggerated. 



220 CRAFTSMANSHIP IN MATHEMATICS 

7. 54 minutes ago, it was 5 times as many minutes past 5 
as it is now minutes to 7. What is the time now? 

Most watch and clock problems can be solved on the basis 
of the principle illustrated in the last example, and one 
careful analysis, to exemplify the method, is usually enough 
to enable the boys to attack successfully most of the problems 
given in a textbook. But this problem, another absurdity 
from an examination paper, does not fit into any general 
scheme. Though easy, its translation into an equation may 
at first puzzle most average pupils. 

The basis for equalizing quantities is pretty obvious at 
the outset: 

(54 min. ago, no. of min. past 5) = 5 (no. of min. to 7 now). . (i) 

The question to be answered is, what is the time nowl 
The problem mentions 5.0 and 7.0, and refers to the time 
now as a number of minutes to seven. 

Hence, let the time now be x minutes to 7. . . . (ii) 

We also require to know what the time was 54 minutes 
ago\ this must have been (x + 54) minutes to 7. 

But we have to express this time in terms of minutes 
past 5.0. Now 5.0 is 120 minutes before 7.0. 

Hence, 54 minutes ago the number of minutes past 5.0 

was 120 (x -f 54) (iii) 

From (i), (iii) is 5 times (ii), 

i.e. 120 - (x -|- 54) = 5*. /. x = 11. 

All answers to equations should be checked; checking in a 
case like this is particularly necessary. 

Since x= 11, the time now is 11 minutes to 7, or 6.49. 
The time 54 minutes ago was 5.55, or the number of minutes 
(55) then past 5.0 is 5 times the number of minutes (11) now 
to 7.0. 



ALGEBRAIC EQUATIONS 221 

8. Three friends going on a railway journey take with 
them luggage amounting in all to 6 cwt. Each has more than 
can be carried free, and the excess charged them is 2s. 6d., 7s., 
and 10s., respectively. Had the whole belonged to one person, 
he would have had to pay 34s. 6d. excess. How much luggage 
is each passenger allowed to carry free, what is the excess 
charge per Ib., and what is the weight carried by each of the 
three friends? 

Consider first a simple case. If I am allowed to take 
with me, say, 100 Ib. free, and have to pay, say, \d. on every 
Ib. exceeding 100, then if I take with me a total of, say, 150 Ib. 
the excess I have to pay is \d. X (150 100). 

Thus a possible form of equation seems to be: 

(excess charge per Ib.) X (no. of Ib. excess) = (total charge for excess), 

and as there are two separate though similar statements 
concerning excess, we ought to be able to formulate two 
equations, say in x and y. 

What have I to find out? (1) Ib. per person carried free, 
and (2) excess charge per Ib. Hence: 

Let each passenger 

(1) Carry x Ib. free. 

(2) Pay y pence on each Ib. excess. 

Assume that one of the friends takes 3 tickets and shows 
them to the porter, who on weighing the luggage and finding it 
to be 672 Ib., deducts 3x from the 672, and charges y pence per 
Ib. on the difference, viz. (672 3#). Since the sum actually 
paid for excess = 2s. 6d. + 7*. + 10s. = 19s. 6d. = 234^., , 

/. (672 - 3x)y = 234 (i) 

But if all 672 Ib. had belonged to one person, he would 
have taken only 1 ticket, and the porter would have charged 
y pence on each of (672 x) Ib. Since the sum actually paid 
in this case for excess = 34s. Qd. = 414J., 

.'. (672 - x)y = 414 (ii) 



222 CRAFTSMANSHIP IN MATHEMATICS 

Dividing (i) by (ii) we have 

672 - 3x _ 13 

672- * 23' /. #= 120 

= no. of Ib. allowed free. 
Hence from (i) or (ii), 

y \d. = charge per Ib. excess. 
Weight in Ibs. carried by each person 

= 120 + ~, 120 + 8 * 120 + 1|?, respectively. 
Ill 

This problem is worth giving a class a second time, say 
three months after the first. Boys seem to find a first analysis 
a little difficult. 

9. An express train and an ordinary train travel from 
London to Carlisle, a distance of 300 miles. The ordinary 
train loses as much time in stoppages as it takes to travel 25 
miles without stopping. The express train loses only three- 
tenths as much time in stoppages as the ordinary train , and it 
also travels 20 miles an hour quicker. The total times of the 
two trains on the journey are in the ratio 26 : 15. Find the 
rate of each train. 

We have to think of distance, speed, and time, and their 
relation d = st. 

The ratio of the total times taken by the two trains is 

given, viz. 26:15 (i) 

Hence if we can express these total times in some other way 
we can formulate our equation. 

Distance: 300 miles. 

Speeds'. Let ordinary train travel x miles an hour. 

Then express train travels x -f 20 miles an hour. 

Qf)f) 

Times: Time taken by ordinary train = f- stoppages (in 

x 

hours) (ii) 

Time taken by express train = - + stoppages (in 

X ~j~ 20 

hours) . . (iii) 



ALGEBRAIC EQUATIONS 223 

We now require to know the amount of time lost over 
stoppages. 

(1) Ordinary train: time lost over stoppages is equal to 
that taken in travelling 25 miles: 

Train travels x miles in 1 hour, 

= 1 mile in - hour 
x 

25 

= 25 miles in hours 
X 

25 

.*. - hours - - time lost over stoppages. 
x 

(2) Express train: time lost over stoppages = -^ that of 
ordinary train, 

3 r 25 , 15 , 

_ of hours = hours. 
10 x 2* 

We can now express (ii) and (iii) in the following forms: 

Time taken by ordinary train = ( . -|- ) hours. . (iv) 

\ x x ' 

rr- t t . / 300 . 15\ , , . 

Time taken by express tram = I -f- J hours. (v) 

\x -f 20 2x' 

Hence we have from (i), (iv), (v), 
300 25 

~x x 26 



15 15 



* + 20 2x 

.'. X = 30 (miles an hour). 

If in " racing " and analogous problems the relation d st 
is kept in view, the necessary analysis is usually quite simple. 

10. What is the price of sheep per 100 when 10 more in 
100 worth lowers the price by 50 per 100? 

We must avoid confusion between (1) the number of 
sheep for 100 and (2) the cost of 100 sheep. 

We can find the number of sheep costing 100 if we 
know the price of 1, and we can find the price of 1 if we know 
the price of 100. 



224 CRAFTSMANSHIP IN MATHEMATICS 

A possible equation seems to be: 

(First no. of sheep for 100) 

= (second no. of sheep for 100) 10. . . (i) 

What have we to find? The price of 100 sheep. 

(1) First price of 100 sheep. Call this #. 

:. 1 sheep costs ^ , 

/. number obtainable for 100 = 122 = 10 ' 000 . . . (ii) 

&L x 

100 

(2) Second price of 100 sheep. This = (# 50). 
/. 1 sheep costs ( * " 50) . 

.-. number obtainable for 100 = 



(# 50) x 50 



100 



(i) shows us how (ii) and (iii) are related, and then we may 
make up our equation. 

- 10, ... * = 250. 



x x 50 

i.e. the price of 100 sheep is 250, or 2, 10*. each, or 100 
worth 40 sheep. 

(If 50 for 100, each costs 2, or cost of 100 = 200, 
i.e. 50 less than before.) 

I have found that even Sixth Form boys are sometimes 
baffled by the analysis of this little problem. 

Problems which are at all unusual in form are always 
worth repeating after an interval. 

Books to consult (on the general technique of teaching algebra): 

1. The Teaching of Algebra, Nunn. 

2. Elements of Algebra, 2 vols., Carson and Smith. 

3. A New Algebra, Barnard and Child. 

4. Algebra, Godfrey and Siddons. 

6. A General Textbook of Elementary Algebra, Chapman. 
6. Elements of Algebra, De Morgan. (A valuable old book. So 
are De Morgan's other books, especially his Arithmetic.) 



ELEMENTARY GEOMETRY 225 

CHAPTER XX 

Elementary Geometry 

Early Work 

Some of the younger generation of teachers have never 
read Euclid, and seem to be totally unacquainted with the 
rigorous logic of the old type of geometry lesson. Not a 
few of the old generation regret the disappearance of Euclid, 
urging that the advantages of the newer work are outbalanced 
by the loss of the advantages of the older. 

The real distinction between the older and the newer 
work is, however, sometimes forgotten. Essentially, Euclid 
wrote a book on logic, using elementary geometry as his 
raw material. The amount of actual geometry, qua geometry, 
which he taught was, relatively speaking, trifling. Boys in 
existing technical schools do ten times as much geometry 
as is found in Euclid. But as an exposition of deductive 
reasoning from an accepted set of first principles, Euclid has 
never been equalled. 

Until forty years ago, Euclid was universally taught in 
secondary schools, but the collective opinion of experts had 
gradually hardened against it, partly because the average 
boy found it difficult, partly because some of its propositions 
were too subtle for schoolboys, partly because its foundations 
were far from being unassailable, and partly because the 
actual geometry it expounded was too slight to be of much 
practical service. 

But the geometry that was substituted for Euclid the 
geometry now exemplified in all the ordinary school text- 
books is still Euclidean geometry, i.e. it is a geometry based, 
in the main, on the same foundations as Euclid. These 
foundations consist of a number of quite arbitrarily chosen 
axioms. Other sets of axioms might be substituted for them, 
and then we should get an entirely new system of geometry 

(B291) 16 



226 CRAFTSMANSHIP IN MATHEMATICS 

of a non-Euclidean character. Reference to such geometry 
will be made in a future chapter. 

In practice, the difference between Euclid and the geo- 
metry now taught is in the choice of working tools. In Euclid, 
the proof of every proposition was ultimately traceable to 
the axioms, and every schoolboy had to substantiate every 
statement he made by referring it to something already proved, 
and this in its turn to something that had gone before, and 
so back to the axioms. In those days the axioms were really 
the working tools. But those axioms were so subtle that 
the boys' confidence in them was entirely misplaced. Now- 
adays, the working tools consist of a small number of funda- 
mental propositions. By means of carefully selected forms 
of practical work, the truth of these propositions is shown 
to beginners to be highly probable, but the formal proofs 
of such propositions are not considered until the boys reach 
the Sixth Form. Examination authorities no longer call for 
the formal proofs at the School Certificate stage. 

These working tools once thoroughly mastered, beginners 
plunge into the heart of the subject and make rapid headway. 
In the old Euclidean days a year or more was spent on these 
propositions and a few others, and at the end of that time 
the average boy had but very vague notions about them, 
though the mathematically-minded boy certainly did seem 
to appreciate the rigour of the reasoning presented to him. 

I find it a little difficult to describe the methods of the 
pre-eminently successful teacher of geometry. The methods 
are not the reflection of any particular book but of the man 
himself. By the gifted teacher who happens to be a sound 
mathematician, a new principle is often illuminated by so 
many side-lights that even the dullard can hardly fail to see 
and understand. Successful teachers of geometry seem to 
be those who have given special attention to the foundations 
of the subject, who possess exceptional ingenuity in making 
things clear, and who at an early stage make use of symmetry 
and of proportion and similarity. 

Those teachers who are not successful are often those 



ELEMENTARY GEOMETRY 227 

who confine their work to the limits of the ordinary text- 
book written for the use of boys; who fail to survey the whole 
geometrical field; who are still unacquainted with, or at all 
events do not teach, the great unifying principles of geometry 
duality, continuity, symmetry, and so forth. The little 
textbooks are all right for the boys, but the teaching of geo- 
metry connotes something outside and beyond such text- 
books, especially the principles underlying the grouping and 
regrouping of the thousand and one facts that the beginner 
necessarily learns as facts more or less isolated. The ac- 
cumulated facts can be given many different settings, each 
setting forming a perfect picture, all the pictures different 
yet closely related. 

Work up to 13 or 14 

As I have said in another place, 5 * the following is an 
expression of authoritative opinion as to the nature of the 

work which it is most advisable to do with young boys: 
/ 

1. The main thing should be to give the boys an in- 
telligent (knowledge of the elementary facts of geometry A 

2. No attempt should be made to develop the suoject 
on rigorously deductive lines, from first principles, though, 
right from the first, /precise reasons for statement^ made 
should be demanded. 

3. Young boys are never happy and are often suspicious 
if they feel they are being asked to prove the obvious, but they 
can follow a fairly long chain of reasoning if the facts are 
clear. 

4. All subtleties should be avoided, and, therefore, proofs 
of propositions concerning angles at a point, parallels, and 
congruent triangles should not be attempted, such proofs 
being a matter for later treatment in the upper Forms. 

5. These main working tools of geometry, angles at a 
point, parallels, and congruent triangles, should be presented 

* Lower and Middle Form Geometry, Preface. 



228 CRAFTSMANSHIP IN MATHEMATICS 

, gJ^ 

in such a way as to $nable the boys to understand them 
clearly and to use and apply them readily. 

6. Young boys can easily understand Pythagoras, ele- 
mentary facts about areas, and the main properties of the 
circle and of polygons; and these facts should be taught. 

7. The simple commensurable treatment of (i) the pro- 
portional division of lines, and (ii) similar triangles, should 
be included in the work to be done at the age of 12 to 14; 
young boys soon become expert in the useful practice of 
writing down equated ratios from similar triangles. 

8. By about the age of 13, a boy ought to be able to write 
out a simple straightforward proof formally and to attack 
easy riders. 

9. Throughout the course, all possible use should be made 
of the boys '( intuitions and of their knowledge of space- 
relations in practical life,) relations in three dimensions as 
well as in two. 

10. Responsible teachers should always express them- 
selves in exact (geometrical language, and should make pre- 
cision and accuracy of statement an essential) part of the 
boys' training. 

11. The boys should be taught how to formulate their 
own definitions, and, under the guidance of the teacher, to 
polish up these definitions as accurately as their knowledge 
at that stage permits; and these definitions should be learnt. 
Definitions should never be provided ready-made. 

12. The boys should be taught to realize exactly what 
properties are implied by each definition, and all other pro- 
perties must be regarded as derivative properties requiring 
proof. 

13. The boys may usefully be given an elementary training 
in the principle that a general figure necessarily retains its 
basic properties even when it becomes more and more 
particularized, but that, as the figure becomes less and less 
general, it acquires more and more properties; and vice 
versa. 

14. In short, clear notions of the all-important principle 



ELEMENTARY GEOMETRY 229 

of continuity should, by the time a boy is about 13, " be in 
his very bones ". 

15. A young boy's natural fondness for puzzles of all 
kinds may often usefully be employed for furthering his 
interest in geometry. 

16. In one respect we have drifted too far away from 
Euclid: boys' knowledge of geometry is too often vague, 
too seldom exact. 

Teachers differ in opinion about the degree of accuracy 
to be demanded in beginners' geometrical drawing. Some 
training in the careful use of instruments is certainly desirable, 
but time should not be wasted over elaborate drawings 
when freehand sketches can be made to serve adequately. 
In Technical schools, accurate drawing with instruments is an 
essential part of much of the pupils' work. Even in Secondary 
schools, where the geometry is necessarily given an academic 
bias, a preliminary training in the careful use of instruments 
serves a useful purpose, but there is no point in making Secon 
dary school boys spend time over elaborate pattern drawings 
and designs. Exercises in accurate work of a more telling 
type may be found in the theorems of Brianchon, Desargues, 
Pascal, and others. As theorems, these are, of course, work 
for the Upper Forms; as geometrical constructions, they 
are useful in the lower Middle Forms, where they may be 
learnt as useful and interesting geometrical facts. 

All pupils should be taught the wisdom of drawing good 
figures for rider solving purposes. 

Boys in Technical schools often have a better all-round 
knowledge of geometry than those in Secondary schools 
because they do more work in three dimensions. Solid 
geometry of a simple kind may with great advantage be 
included in the early stages of any geometry course. 

A short course on simple projection at about the age of 
13 helps later geometry enormously. Boys soon pick up the 
main principles, and the work helps greatly to develop their 
geometrical imagination. So does simple work with the 
polyhedra, work which always appeals to boys. 



230 CRAFTSMANSHIP IN MATHEMATICS 

Let logic of the strictly formal kind wait until foundations 
are well and truly laid. The increasing difficulty felt by 
beginners in geometry is largely an affair of increasing diffi- 
culty of logic, and thus we now recognize that parts of the 
third, fourth, and sixth books of Euclid are easier than parts 
of the first book. 

Push ahead. Do not paddle about year after year in the 
little geometrical pond where examiners fish for their questions. 
Even for the examination day such paddling most certainly 
does not pay. 

It is convenient, though not defensible, to preserve the 
old distinction between axioms, postulates, and definitions. 
But if any teacher still believes that axioms should be con- 
sidered in a beginner's course of geometry, let him consult 
Mr. Bertrand Russell, and he will soon be disabused. 

Early Lessons 

Here are a few early lessons in geometry, selected at 
random from the book already cited. The sections have 
been renumbered, seriatim, for convenience of reference, but 
actually the lessons are drawn from all parts of the book. 

LESSON I 
Planes and Perpendiculars 

1. Carpenters, bricklayers, blacksmiths, and plumbers, all 
have to know something about geometry. Architects, builders, 
surveyors, and engineers have to know a great deal about it. 
All of them have to know how to measure things, and how 
to make things perfectly level, perfectly upright, perfectly 
square, perfectly " true "; and much more besides. 

You have already learnt how to use a ruler or scale, 
marked with inches and parts of an inch on the one edge, 
and with centimetres and parts of a centimetre on the other. 
Note that there are very nearly, but not exactly, 2J centi- 
metres to an inch. 



ELEMENTARY GEOMETRY 



231 



You know already that, when two lines meet at a point, 
they form an angle. Here are three angles. 



blade 



Fig. 63 



The middle one is the angle you know best. It is the angle 
you see at the corner of an ordinary picture-frame, or of a 
door, or of a window-frame, or of a table-top. Such an angle 
is called a square angle, or right angle. 

2. You have probably seen a carpenter planing a piece 
of wood, perhaps for a shelf. He begins 

by planing one of the " faces " of the wood, 
and, as soon as he thinks that the face is a 
true plane, he tests it. To do this he uses 
a try -square, which consists of a steel 
blade with parallel edges perfectly straight, 
fixed at right angles into a wooden stock. 
(A /ry-square is a tatf-square.) He holds the stock in his 
hand, and to the planed face of the wood he applies the 
outside edge of the blade, " trying " it in many places and in 
different directions, along and across. If he can see day- 
light anywhere between the blade and the wood, he knows 
that the planed face is not yet a true plane, and that he 
must continue his planing. When it is true, he marks it 
face- side. 

3. Now he turns up the wood so that 
an edge rests on the bench, and he planes 
the edge at the top. Not only has he to 
make this face-edge (as it is called) a true 
plane like the face-side, but he has to make 
the two planes at right angles to each other, 

or, as we usually say, perpendicular to each other. The 
carpenter is not satisfied until the inside right angle of his 
try-square fits exactly, at the same time, the face-side and 



Fig. 64 



232 CRAFTSMANSHIP IN MATHEMATICS 

the face-edge, the blade fitting against the one, the stock 
against the other, the test being made at several places along 
the wood. 

You might use a big try-square to see if a flag-staff or a 
telegraph-post is perpendicular to the ground. If it did not 
fit exactly in the angle between the post and the' ground, 
no matter where tested round the post, you would know 
that either the ground is not level or the post is not upright. 
Two planes, or two lines, or a line and a plane, are 
perpendicular to each other if and only if they are 
at right angles to each other. A perfectly upright post 
in a sloping bank is not perpendicular to the bank because 
it does not make right angles with the bank. 

4. The maker of a try-square guarantees the accuracy 
of the inside angle, but not of the outside right angle. Thus 
you may use it for testing the right angles 
of a table-top, of a door, of the outside of 
a box. It is not advisable to use it for 
testing the inside right angles of a box, or 




i^g. 6 S of a drawer, or of a door-frame. (Strictly 

speaking, it ought not to have been used 
for testing the right angle round the flag-staff or telegraph- 
post.) For testing inside right angles, we use an architect's 
set-square, a flat triangular piece of wood with a true right 
angle. You will be given two of these to work with, a little 
later on. 

5. You have learnt that, when you want to find out if a 
surface is a true plane, you must test it with an accurately 
made straight-edge of some kind. If, on the other hand, 
you are doubtful about the accuracy of a straight-edge (an 
ordinary ruler, for instance), you can test it by applying 
it to a plane known to be true. Thus, a true straight-edge 
may be used for testing a plane, and a true plane may be 
used for testing a straight-edge. One must be true, and then 
it may be used for testing the other. 

(When numbered statements in dark type are followed by the 
letter " L ", the statements are to be learnt, perfectly.) 



ELEMENTARY GEOMETRY 233 

6. A PLANE SURFACE (or a PLANE) is a surface 
in which a true straight-edge will everywhere fit 
exactly. (L.) 

LESSON II 

Horizontal, Vertical, and Oblique Lines and Planes 

7. Borrow a spirit-level from the Geography Master, 
or from the school carpenter, and see if the floor of your room, 
the top of the table, the window ledge, and the mantelpiece 
are horizontal (perfectly level). (Your master will explain 
how the spirit-level is made and used.) If a plane surface 
is everywhere horizontal, the surface is called a horizontal 
plane, and straight lines drawn on that surface are horizontal 
lines. The surface of still water (in a basin, for instance) is 
a horizontal plane, and floating lead-pencils may be regarded as 
representing horizontal lines. The edge of a book-shelf, the 
edge of a table- top, the joints of floorboards, the line where the 
floor meets a wall, are other examples of horizontal lines. 

8. You have probably seen a bricklayer use a plumb- 
line a cord stretched straight by a hanging leaden weight. 
He uses it to see if the walls he is building are vertical 
(perfectly upright). Make a plumb-line for yourself, and see 
if your school walls are vertical. If they are vertical and if 
they are plane, their surfaces are vertical planes. Cover 
the plumb-line with chalk, hold it close to the wall and let 
it come to rest, then pull it out towards you a little way and 
let it go suddenly. It springs back and leaves a straight 
chalk-line on the wall. This straight line is a vertical line. 
The balusters on a stair-case, hanging chains, hanging 
ropes, telegraph poles, the lines where any two walls of a 
room meet each other, may all be regarded as representing 
vertical lines. Rain-drops fall in vertical lines, unless there 
is a wind. A telegraph post fixed in horizontal ground is 
both vertical and perpendicular; if fixed in a sloping bank, 
it is vertical but not perpendicular. Why? 

Vertical lines always point downwards, towards the 
centre of the earth. 



234 CRAFTSMANSHIP IN MATHEMATICS 

Butterflies alight with their wings in vertical planes, 
moths with their wings in horizontal planes. 

9. Planes and straight lines which are neither vertical 
nor horizontal are called oblique. Oblique means slanting 
or sloping. 

10. On the vertical surface of a wall, it is easy enough to 
draw both horizontal and vertical and oblique lines, but 
vertical and oblique lines cannot be drawn on a horizontal 
sheet of paper lying on the table. All lines on a horizontal 
plane are horizontal. Yet it would be inconvenient to have 
to draw lines on a sheet of paper which is pinned to the 
wall, though sometimes your master certainly does draw 
lines on a vertical blackboard. It has been decided, just as 
a matter of convenience when drawing, to represent the 
three different kinds of straight lines all on a horizontal 
plane, and in this way: horizontal lines, parallel to the top 
and bottom edges of your paper; vertical lines, parallel 
to the left- and right-hand edges of your paper; oblique 
lines, lines in any other direction. 




Horizontal lines Vertical lines Oblique lines 

But remember that, as long as your paper is lying on the 
horizontal table, it is not strictly true to say that the lines 
you draw on it are anything but horizontal. We do not 
obtain a true picture unless we hold the paper in a vertical 
plane (against the wall, for instance). Then the vertical 
lines may be made to appear really vertical. 

11. HORIZONTAL PLANES are planes which are 
perfectly level. (L.) 

12. HORIZONTAL LINES are straight lines in a 
horizontal plane. (L.) 

13. VERTICAL PLANES are planes which are 
perfectly upright. (L.) 



ELEMENTARY GEOMETRY 



235 




Fig. 66 



14. VERTICAL LINES are straight lines in a 
vertical plane that point downwards towards the 
centre of the earth. (L.) 

15. Planes and straight lines which are neither 
horizontal nor vertical are called OBLIQUE. (L.) 

LESSON III 
Solids and Surfaces 

16. Here is a brick. Measure it. It is 9" long, 41" broad, 
3" thick. It is a solid body, but, in geometry, we call it a 
solid not because it is made throughout of a 
particular kind of hard stuff but because it 

occupies a certain amount of space. If the 

brick were hollow and made of paper, we should 

still call it, in our geometry lessons, a solid. 

A room of a house is a solid; so is an empty box. Both have 

length, breadth, and thickness. But we do not usually 

speak of the thickness of a house or of a box. We say that a 

house has length, breadth, and height, and a box length, 

breadth, and depth. But all have three dimensions; that is, 

we can measure them from front to back, from side to side, 

and from top to bottom. (Both the word dimension and the 

word mensuration are derived from 

the same Latin word, mensura, a 

measure.) 

17. Here are a cube (fig. 67, i) 
and a square prism (ii). You 
have probably seen them before, 
and know their 

names. If they 
were made of paper, 
we could run a 
knife along some of 
their edges and lay 
them out flat like 

this: Fig. 68 





Fig. 67 




236 



CRAFTSMANSHIP IN MATHEMATICS 



Plans of this kind are called the nets of the solids. Later 
on in the geometry book, you will find instructions how to cut 
out nets from stiff paper and how to fold and bind them 
up into the solids they represent. 

18. The surface of both the cube and the prism consists 
of six faces. All six faces of the cube are squares. Only 
two faces of the prism are squares, the other four being 
oblongs. Sometimes we speak of the two square faces of 
the prism as ends or bases, and the four oblong faces as 
the sides. In each case, all the faces are, of course, planes. 
Any two adjoining planes of the cube or of the prism meet 
in an edge, or, as we sometimes say in geometry, the two 
planes intersect in a straight line. 

19. Here are four more solids which you have probably 
seen before; a square pyramid, a cylinder, a cone, and 
a sphere. 






Fig. 69 



20. In shape, the square pyramid reminds you of the 
famous Egyptian pyramid. Its surface consists of five plane 
faces, namely, one square base, and four triangular faces 
meeting in a point called the vertex. The vertex is exactly 
over the centre of the base * (fig. 69, i). 

21. The cylinder reminds you of a garden roller, of a 
jam-jar, or of part of a pipe or tube. You can imagine it 
spinning on an axis. When rolled on the ground it runs 
in a straight line. The complete surface of the cylinder 
consists of two circular plane surfaces separated by a curved 
surface (fig. 69, ii). 

22. The cone reminds you of the old-fashioned candle- 



* It is convenient to be able to refer to the " centre " of a square, but it is not 
strictly correct. A circle has a true centre, so has a sphere. 



ELEMENTARY GEOMETRY 



237 



extinguisher, or of the sugar loaf. It has a circular base, 
and it is so far like a pyramid that it has a vertex over the 
centre of the base. You can imagine it spinning on an axis. 
When rolled on the ground it runs round in a circle. The 
complete surface of the cone consists of one circular plane 
surface and a curved surface (fig. 69, iii). 

23. The sphere reminds you of a ball of some kind, 
and it is a ball which is perfect in this way the point called 
the centre is exactly the same distance from every point 
on the surface. You can imagine it spinning on an axis, 
like the earth. When rolled on the ground, it will run in 
any direction. The surface of a sphere is everywhere curved 
(fig. 69, iv). 

24. When we speak of a " cylindrical surface " or of a 
" conical surface ", we usually refer to only the curved surface 
of the cylinder or cone. It is important to notice that this 
curved surface of the cylinder and the cone is very different 
from the curved surface of a sphere. If you place a sphere 
upon a plane (say a table), it touches the plane in a point. 
If you allow a cylinder or a cone to lie with its curved surface 
on a plane, it touches the plane in a line. 

25. We can make nets of a cylinder and a cone, but not 
of the sphere. Here arc nets of a square pyramid, a cylinder, 




and a cone, but to make models from the nets of the last 
two is a little difficult. 

26. You have learnt to recognize a square, an oblong, 
a triangle, and a circle. All these are called plane figures, 



238 CRAFTSMANSHIP IN MATHEMATICS 

because each encloses, within a boundary line, part of a 
plane surface. All plane figures have closed boundary 
lines. The letter O and the letter D are geometrical figures, 
but not the letter C or the letter W. Straight-lined figures 
like squares and triangles are called rectilineal figures. 
A circle is a curved figure. 

27. A PLANE FIGURE is part of a Plane, and it 
is separated from the rest of the Plane by a boundary 
line. (L.) 

28. A PLANE RECTILINEAL FIGURE is a straight- 
lined figure on a Plane. (L.) 

29. RECTI-LINEAL means straight-lined. (L.) 

LESSON IV 
Angles 

30. If I stand facing the east and the drill sergeant says 
" left turn ", I turn and face the north, and I have then 

turned through a right angle. If he repeats 
the order, I turn to the west, and I have then 
turned through another right angle. If he 
repeats the order twice more, I turn and face 
south and then turn and face east, by which 
time I shall have turned through four right 
angles. I have made one complete rotation 
(Lat. rota = a wheel). Note the little arrow showing my 
first quarter-rotation or right angle. 

Evidently an angle may be smaller or greater than a right 
angle. Whenever you look at a clock, the two hands are 
making an angle with each other. In fact, they are making 
new angles with each other all day long. Even when they 
are exactly together they have just completed a new angle 
and are just beginning to make others. 

31. On paper an angle is represented by two lines meeting 
in a point. The two lines are called the arms of the angle, 
and the point where the two arms meet is called the vertex 



w 



ELEMENTARY GEOMETRY 239 

of the angle. The same angle may have long arms or short 
arms. If a big clock and a little watch are both keeping 
correct time, the angles between their hands are always 
exactly the same. An angle always represents an amount 
of movement, namely, the movement of rotation. One 
arm shows where the rotation began, and the other where it 
finished, and you must always think of an angle in this way. 

32. You can measure angles of different sizes fairly well 
by opening and closing your dividers, but the joint prevents 
you from making an angle of a whole rotation. A more 
convenient form of angle -measurer is necessary, and you 
may make one in this way. Take two nicely planed strips 
of wood, say about 12" long, |" wide, -J" thick, and pivot 
them together, something like your dividers, by means of 
a tiny brass bolt with rounded head and nut, generally 
obtainable for a penny or two from the ironmonger's. If 
you cannot obtain these things, two strips of cardboard 
will do, pivoted on a long drawing-pin, head downwards, 
with a protecting bit of cork over the point. 

Place your angle-measurer on the table before you, the 
vertex O to the left, the two arms OA, OB together as if 



A,B 



fcj 

.L, 



Fig. 72 




they were both pointing to III on a clock-face (fig. 72, i). 
Keep the under arm OA fixed, and rotate the upper arm 
OB. Rotate it in an anti-clockwise direction (this is the custom 
in geometry), and make angles equal to one, two, three, and 
four right angles. Draw the four angles, and in each case 
show the amount of rotation by means of little curved arrows 
(fig. 72, ii, iii, iv, v). 

33. In measuring different quantities, weights and mea- 
sures for instance, big units like tons and miles are some- 




240 CRAFTSMANSHIP IN MATHEMATICS 

times inconvenient. We do not weigh our tea in tons or 
measure our pencils in miles; we use smaller units like 
ounces and inches. So with angles. A right angle is a rather 
00* big unit, and sometimes we use a 

smaller unit called a degree. If we 
make a right angle as before, but move 
OB into position gradually, in ninety 
equal steps, each of these steps is an 
angle of one degree. It is a very 
small angle, too small to be shown 
clearly on paper unless we make the 
n g , 73 arms very long. The figure shows that 

even an angle of 5 degrees is very small. 
The sign for " degree " is a little circle placed at the top 
right-hand corner of the given number. Thus for " 35 
degrees " we write, " 35 ". 

34. We may make up a little table: 

90 degrees make a right angle, 

2 right angles make a straight angle, 

2 straight angles make a perigon. 

A perigon is an angle of one complete rotation (pert round; 
gon ~ angle). It is equal to four right angles, or 360. A 
straight angle contains 180 ( 32, fig. 72, iii). 

We choose the number 360 for the perigon simply because 
it is the number which contains many useful factors (2, 3, 
4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 
90, 120, 180). Any other number would do, but it would be 
less useful. The French use the number 400; they prefer 
to divide up the right angle into 100 degrees (they call them 
grades) instead of 90. 

35. Note the number of degrees in the angles of fig. 74. 
The dotted lines show the right angles, and help the eye 
to estimate the numbers of degrees. 

Practise drawing angles of different sizes, and estimating 
the number of degrees. The most important angles of all 
are 30, 45, 60, 90, 180. The easiest to make is, of course 



ELEMENTARY GEOMETRY 



241 



an angle of 90. Divide it into two equal parts as accurately 
as you can, and so obtain 45. It is pretty easy to divide, 




iB 



Fig. 74 



with fair accuracy, 45 into three equal parts, in order to 
obtain 15 and 30. And so on. 

But guesswork will certainly not always do. It is often 
necessary to draw given angles accurately, and for this pur- 
pose you must use a pro- 
tractor. A protractor is 
a semicircular * piece of 
brass or celluloid, with 
numbers from to 180 
round the circumference, 
in both directions, and by 
means of it you can make Fig. 75 

an angle of any size. 

Suppose you have to draw a line, at say, 55 with a given 
line. Place the straight diameter of your protractor against 
the given line, in such a way that the marked midpoint 




*A surveyor's protractor is circular, and is numbered up t 
geometry we do not very often require angles greater than ioo. 

(B291) 



to 360. But in 
17 



242 CRAFTSMANSHIP IN MATHEMATICS 

of the diameter is against that end of the line that is 
to be the vertex of the angle. At the number 55 on the 
circumference, mark a point on the paper. Remove the 
protractor, and through that point draw the second arm of 
the angle. 

But, you will say, there are two 55's on the circumference. 
How are we to choose between them? That is easy, for 
you know that 55 is less than a right angle, and you choose 
the 55 which will give you such an angle. The other 55 
would be used if the vertex of the angle had to be at the 
other end of the line. 

36. When two straight lines stretch out from one 
point, like two spokes from the hub of a wheel, they 
form an angle. (L.) 

37. The two lines are called the arms of the angle, 
and the point where they meet is called the vertex 
of the angle. (L.) 

38. An angle always shows a certain amount of 
ROTATION round the vertex, one arm showing 
where the rotation began, the other arm showing where 
it ended. (L.) 

39. A PERIGON is an angle of one complete 
rotation. (L.) 

40. A STRAIGHT ANGLE is an angle of a half 
rotation. (L.) 

41. A RIGHT ANGLE is an angle of a quarter 
rotation. (L.) 

42. AN ANGLE OF ONE DEGREE is an angle of 
sio P art * a rotation. 

LESSON V 
Surveyors and their Work 

43. A surveyor's work is to measure up land, and to 
draw plans and maps. For measuring lengths, he uses a 
long chain of 100 links. For measuring angles, he uses an 




ELEMENTARY GEOMETRY 243 

angle-measurer which is like yours in this respect that 
it consists of two pivoted arms; but it is much more elaborate 
than yours, for he has to measure angles very accurately. 
He also uses a levelling-staff, to help him measure differences 
of level. A levelling-staff is merely a pole, graduated to show 
heights above the ground. 

44. Here is a problem in which the necessary angle 
measurements may be correctly and easily made with one 
of your set-squares. To solve it you must make a drawing 
to scale. 

A and B are two towns 20 miles apart. Another town C 
is 60 east of north from A and 30 
west of north from B. Draw a plan to 
show the position of C, and give its 
distances from A and B. 

The line AB is 20 miles long, 
and we have to draw it to a suitable 
scale. A scale of -|" to the mile would 
do. Thus we make AB 20 eighth-inches, or 2|", long. 

If C were exactly north of A, it would be somewhere 
in the line AD. But it is 60 east of this line, and we there- 
fore make the angle DAF equal to 60. Again, if C were 
exactly north of B, it would be somewhere in the line BE. 
But it is 30 west of this line, and we therefore make the 
angle EBG equal to 30. We know now that the town C 
lies on both AF and BG. But the only place where it can 
lie on both is where they meet. Hence, mark this point, C. 
We have thus found the position of C. 

To find the distances CA and CB, we measure them to 
scale. CA is nearly 17J eighth-inches long, and CB is 10 
eighth-inches long. Thus C is 17 miles from A and 10 
miles from B. 

But this problem was a problem on paper. No part of 
the work was done with measuring instruments in the 
field. Let us come back to the surveyor. 

Sometimes a surveyor works on level ground, and has to 
measure angles in a horizontal plane. Sometimes he works 



244 



CRAFTSMANSHIP IN MATHEMATICS 



on hilly ground and has to measure angles in a vertical plane. 

45. Measuring an angle in a horizontal plane. 

Suppose you are standing at a place P in a field, and you 
imagine a line drawn from your eye to each of two distant 
trees, T l and T 2 . What is the angle between the lines? Set 
up a table at P, with a piece of drawing-paper pinned on 
it. (A camera tripod stand with a drawing-board fixed on it 
horizontally about the height of your top waistcoat button 
would do nicely.) Place an angle-measurer on the table, 
swing one arm round to point to T x , and the other round 
to point to T 2 . Hold the arms firmly and draw the two 
angle lines (against the inside of the arms), remove the measurer, 
and with your protractor find the number of degrees in the 
angle. 

(Ask the geography master to show you his plane-table 
and to explain how he measures angles made by distant 
objects. With his angle-measurer pivoted to the centre of a 
circular protractor on the table, he is able to read at once 
any angle made by the two arms.) 

46. Measuring an angle in a vertical plane. Pivot 
your angle-measurer to the side of a short post, or to the 

side of a stout stick thrust 
vertically into the ground, in 
order that the arms may swing 
in a vertical plane. An ordi- 
nary drawing-pin makes a poor 
pivot, for it is then difficult to 
make the arms remain in a 
particular position. An angle- 
measurer made of wood, with 
a fairly tight wooden or metal 
pivot, is much more satisfactory 
than the pivoted cardboard 
strips. An angle measured 

in a vertical plane is always an angle with a hori- 
zontal arm; the other arm points upwards or downwards 
as may be necessary. 





Angle of elevation Angle of depression 
Fig. 77 



ELEMENTARY GEOMETRY 245 

If you are on low ground and want to measure the angle 
made by, say, a cottage at the top of the hill, point the one 
arm of your angle-measurer upwards to the cottage, and 
measure the angle of elevation. If you are on high ground, 
say the top of a cliff, and want to measure the angle made 
by a boat in the water below, point the one arm of your 
angle-measurer downwards to the boat, and measure the 
angle of depression. Since, for measuring different angles, 
the arms may have to swing round in different vertical planes, 
it is an advantage to be able to turn the post round in the 
ground, and it should therefore have a rounded point, some- 
thing like the point of a cricket-stump, prolonged. 

(Ask the geography master to show you his clinometer, and 
to explain how he reads, from the cardboard protractor, 
angles of elevation and depression. Try to understand the 
use of the little plumb-line, and observe the pivot on which 
the protractor turns.) 

47. How can I find the width of a river which I 
cannot cross? To solve this problem we have to measure 
angles in a horizontal plane. E 

Let AB and CD repre- A ~ 

sent the two banks. I note 

some object E on the oppo- C 

site bank, and I measure Fig. 78 

any length FG, say 100 

yards, on the near bank. Then I measure the horizontal 
angles at F and G, in each case pointing one arm of my angle- 
measurer along CD and the other arm to the object E. I 
note that angle EFG = 60, and angle EGF = 45. Now I 
am ready to make a drawing to scale. A scale of 1" to 
50 yards seems convenient, so that FG = 2". The width 
of the river is shown by a perpendicular EH drawn from 
E to FG. Measuring EH to scale, I find it is very nearly 
yards. 

48. How can I find the height of the flagstaff in the 
school field? To solve this problem I have to measure an 
angle in a vertical plane, and as one arm of my angle-measurer 



246 



CRAFTSMANSHIP IN MATHEMATICS 



will have to point upwards, the angle will be an angle of 
elevation. 

Let AB be the flagstaff, and let CD be the post to which 
my angle-measurer is attached: a convenient height of this 
attachment is 4' above the ground. The post may be fixed 
at any measured distance from the 
flagstaff, say 20'. Thus DB = 20'. 
The horizontal arm of the angle- 
measurer points to E in the flagstaff; 
E is therefore 4' above the ground. 
The other arm points to the top of 
the flagstaff. I now measure the angle 
ACE, and find it is 60. Now I am 
ready to make a drawing to scale, say 
1" to 10', so that CE (= DB) = 2", 
angle ACE - 60, angle AEC = 90. 
The length of AE, measured to scale, 
is 34-6'. Hence AB = 34-6' + 4' 
= 38-6'. 

(The length BE is exaggerated in 
the printed figure.) 

49. The next time you see a surveyor at work, ask him 
to show you the instruments he uses for measuring horizontal 
and vertical angles, and to explain how he is able to measure 
even very small fractions of a degree. Also ask him to tell 
you something about his levelling-staff and his chain. 




LESSON VI 
Symmetry 

50. Stand in front of a looking-glass, with a book (or 
some other object) in your right hand. In the glass you see 
an image of yourself, but the image holds the book in his left 
hand. Close your left eye; the image closes his right eye. 

Hold open your right hand in front of the glass, and look 
at the image of the palm. Compare this image with the 



ELEMENTARY GEOMETRY 247 

palm of your real left hand. They are exactly alike. For 
instance, the two thumbs point in the same direction. 

Thus the image of your right hand is a left hand. 
In short, your two hands are not in all respects alike; each 
is the " image " of the other. 

51. Place a pair of gloves side by side on the table, 
backs upwards, thumbs touching. Each is the image of the 
other. Turn the left glove inside out; it has become a right- 
hand glove. You now have two right-hand gloves, no longer 
images of each other but like each other. 

52. Fold a sheet of white paper, like a sheet of note- 
paper, and smooth down the crease. Open again, and let 
a drop of ink fall in the crease. Now fold, and press the 
folded paper fairly hard, to make the ink run and form a 
pattern. Open; the two half-patterns are right- and left- 
handed; each is the image of the other. When the paper 
is folded on its crease and held up to the light, the two half- 
patterns are seen to fit over each other exactly. 

53. Right- and left-handed patterns that can be folded 
exactly together in this way, and are thus images of each 
other, are said to be symmetrical. The dividing line re- 
presented by the crease is called the axis of symmetry. 
We say that the doubled pattern is symmetrical with respect 
to the axis. 

54. Take another sheet of paper, and fold as before. 
Let a drop of ink fall inside, but at some distance from the 
crease. Press down the doubled paper, and 

so form ink figures. The figures are images of 
each other and fold together exactly as before, 
but this time they do not touch the crease (the 
axis). That does not matter. They are still 
symmetrical with respect to the axis. 

55. In the accompanying figure (a kite), 

the line AB is evidently an axis of symmetry, FI. so 

for the half ACB can be folded over on AB 
and be made to fit exactly on the other half. The one half 
is the image of the other. Hence corresponding lines on 






248 CRAFTSMANSHIP IN MATHEMATICS 

the left and right must be equal in length. If we hold 
the doubled paper up to the light, we can see that the lines 
are equal. Corresponding angles must also be equal. 

56. ABC and DBF are two figures symmetrical with 
respect to the axis MN. Hence, if we fold on MN, the 

figures will fit together exactly. On 
ABC, mark the two points G and 
H, fold over, and prick through G 
and H, on DEF. On opening out 
we shall find the two points K and 
L in positions corresponding ex- 
actly to G and H, and KL is 
evidently equal to GH. 

57. Every point, every line, every 
Fig. si angle, in one of two symmetrical 

figures has an image in the other. 

The image always corresponds exactly to the original. The 
two have exactly corresponding positions. 

58. Thus a point and its image are always equidistant 

from the axis. In the last figure, for instance, A and D are 

equidistant from MN, for they come together when the 

figures are folded about the axis. Hence if we join AD, the 

axis must bisect AD. 

59. Fold a piece of paper and mark the 
crease as an axis AB. On one side of the axis, 
make a point M. Fold, and prick through M 
to obtain its image N. Join MN. PM = PN 
(by 58). Angle APM = angle APN (by 
57). But angle MPN is a straight angle, and 
thus the equal angles APM and APN are both 
right angles. Also, the angles vertically opposite 
Fig. 82 these are equal. Hence all four angles at P are 
right angles. We see now that the axis not 
only bisects MN but is perpendicular to it, that is, the axis 
is the perpendicular bisector of MN. Observe that 
whenever you fold a sheet of paper a second time, as when 
you put it into an envelope, you make two axes of symmetry 



M 



ELEMENTARY GEOMETRY 



249 



perpendicular to each other, the four perfect right angles 
fitting exactly together in the envelope. When the paper 
is opened out, you see the complete perigon they form. 

60. Fold a piece of paper, and then fold a second time, 
thus obtaining two axes of symmetry, MN and PQ, and 
four right angles. The four divisions are sometimes called 
quadrants. Prick through all four thicknesses of the folded 
paper, in four or five points not in the same straight line. 
Open out, and join up the 

points, in the same manner, in 
the four quadrants, thus mak- 
ing four figures. Convince 
yourself that both MN and PQ 
are really axes, by first folding 
on MN, holding up to the light 
and seeing that the left and 
right halves of the whole fit, 
then folding on PQ and seeing 
that the upper and lower halves 

fit. Join any point, say A, to its image on the other side 
of each axis, namely to B on the other side of MN, and to 
C on the other side of PQ. Observe that the axes are the 
perpendicular bisectors of the respective joining lines, MN 
of AB, and PQ of AC. So it is generally. 

61. When two figures are symmetrical with respect 
to an axis, they are right- and left-handed, and when 
they are folded about the axis, they fit together exactly. 
(L.) 

62. An axis of symmetry is the perpendicular 
bisector of the line joining any point on one side of 
the axis to its image on the other side. 




250 CRAFTSMANSHIP IN MATHEMATICS 

LESSON VII 
Congruent, Symmetrical, Similar 

63. We often require a word to describe two figures 
which are alike in all respects corresponding lines the 
same length, corresponding angles the same, areas the same, 
appearances the same. 

When two figures are exactly alike in all respects, and 
can be made to fit exactly together, they are said to be con- 
gruent. (Congruent means exactly agreeing.) Here are 
three pairs of congruent figures. 




Fig. 84 

64. Symmetrical figures are exactly alike in all respects 
save one: they are right- and left-handed. To make two 
symmetrical figures fit exactly together, we have to turn 
one of them over through a straight angle (180), round 
the axis of symmetry. It is like picking one up, turning it 
upside down, and putting it down again. Then the two 
will fit exactly. 

Strictly speaking, we ought not to call symmetrical 
figures congruent, because they are not alike in all respects; 
they are right- and left-handed. But it has become customary 
to call even symmetrical figures congruent, because they can 
be made to fit exactly if one is turned over. 




i. Congruent ii. Symmetrical iii. Similar 

(but not symmetrical) (and congruent) (neither congruent nor 

symmetrical) 
Fig. 85 

65. But similar is another term altogether. Similar 



ELEMENTARY GEOMETRY 251 

figures are figures of the same appearance, irrespective 
of their size. (See fig. 85.) 

66. Congruent figures which are not symmetrical may 
be made to fit together exactly by sliding one over the other. 
But symmetrical figures cannot be made to fit by sliding; 
one has first to be turned over. 

You might say that since congruent figures are alike 
in appearance, we might call them similar. That is true, 
but in geometry we do not usually apply the term similar 
to congruent figures unless they are of different sizes. 

67. Notice two important things about similar 
figures: (1) all the angles in the one are equal to the 
corresponding angles in the other; (2) the proportions in 
the one are equal to the proportions in the other. (If, for 
instance, the big pig's tail is one-third the length of his 
back, the little pig's tail is one-third the length of his back.) 
You will learn more about " proportions " later on. 

68. CONGRUENT figures are figures exactly alike 
in all respects. One can be made to slide over the 
other and fit. (L.) 

69. SYMMETRICAL figures are right- and left- 
handed congruent figures. To make them fit, one has 
to be turned over through 180, (L.) 

70. SIMILAR figures are figures of different sizes, 
but they have the same appearance, the same pro- 
portions, and the same angles. (L.) 

LESSON VIII 
Classifying and Defining 

71. When we arrange a number of things in separate 
classes, we are said to classify them. 

We may, for instance, arrange all school exercise-books 
in two quite distinct classes, namely, ruled and unruled. 
Such a classification is good. But suppose we say that 
all the people in London are either males, or females, or 



252 CRAFTSMANSHIP IN MATHEMATICS 

Australians. The classification is bad, for the Australians have 
been included twice over; they are all males or females. 

Here is another example of a good classification. In a 
certain school, the 100 boys in Form IV are grouped in 
four divisions, according to the languages they learn in 
addition to English and French. 

Form IVa learn both Latin and Greek, but not German. 
Form IVb learn both Latin and German, but not Greek. 
Form IVc learn Latin, but not Greek or German. 
Form IVd learn German, but not Latin or Greek. 

There are four distinct divisions. Every boy is included 
once, and only once. 

72. Now we will classify triangles. All triangles are 
either isosceles or scalene. But isosceles triangles are of two 
kinds, those with two sides equal, those with all three sides 
equal. Thus we may arrange the classes in this way: 

Triangles 



no two sides equal at least two sides equal 

I 

I I 

only two sides equal all three sides equal 
or we may arrange in this way: 
Triangles 



scalene isosceles 



I I 

base shorter or longer base equal to the equal sides 
than the equal sides equilateral 

73. Defining. A thing (it may be a dog or it may 
be a triangle) has a name, and that name is a word. In 
order to say what that word means, we have to make a 
short statement which will show how the thing is dis- 



ELEMENTARY GEOMETRY 253 

tinguished from all other things. That short statement is a 
definition. We define a word, and the definition must 
include the leading property of the thing. 

We begin by thinking of the class to which the thing 
belongs. Suppose, for instance, we have to define a chair. 
To what class of things does a chair belong? Evidently 
to the class articles of furniture. Thus we may begin 
by saying, 

A chair is an article of furniture . . . 

Now we have to pick out the particular property which 
distinguishes a chair from all other articles of furniture. 
What is a chair specially used for? For sitting on. Thus 
we may now say, 

A chair is an article of furniture for sitting on. 

But benches, sofas, and stools are also used for sitting on. 
How are we to distinguish a chair from these? Benches 
and sofas are made for more than one person to sit on. So 
we may say, 

A chair is an article of furniture for one person to sit on. 

But this might apply to a stool. How are we to distinguish? 
A chair has a back, a stool has not. We therefore say, 

A chair is an article of furniture for one person to sit on 
and to lean back against. 

Again, define a pair of compasses. To what class 
does it belong? Mathematical instruments. What is its special 
use? For drawing circles. Thus we make up the definition: 

A pair of compasses is a mathematical instrument for drawing 
circles. 

74. Define a triangle. 

To what class does it belong? Plane rectilineal figures. 
What property distinguishes it from all other plane 
rectilineal figures? It has three sides. Therefore we say, 

A triangle is a plane rectilineal figure with three sides. 



254 CRAFTSMANSHIP IN MATHEMATICS 

Define an isosceles triangle. 

To what class does it belong? Triangles. 

What distinguishes isosceles triangles from the other 
great class of triangles (scalene)? Equality of the two sides 
from the vertex to the base. 

Therefore we say, 

An isosceles triangle is a triangle in which the two sides 
from the vertex to the base are equal. 

Define an equilateral triangle. 

To what class does it. belong? Isosceles triangles. 
What distinguishes it from other isosceles triangles? The 
base is equal to each of the other two sides. 
Therefore we say, 

An equilateral triangle is an isosceles triangle in which 
the base is equal to each of the other two sides. 

Another definition of an equilateral triangle is some- 
times given: an equilateral triangle is a triangle with three 
equal sides. But this definition is not so good as the other. 

75. We might, if we liked, classify triangles according 
to their angles, and ignore their sides. The sum of the three 
angles of a triangle is 180. Hence, if a triangle has an obtuse 
angle, the other two angles must be acute; or if it has a right 
angle, the other two angles must be acute; if it has neither 
an obtuse angle nor a right angle, all three angles must be 
acute. Thus we have a new classification: All triangles 
are either obtuse-angled triangles, or right-angled 
triangles, or acute -angled triangles. 

But do not mix up the two classifications of triangles. 
That would take us back to the Australians! 

Such lessons are easily within the range of very young 
boys. 

Some teachers are, however, curiously afraid of the prin- 
ciple of symmetry, urging that it does not lend itself to 



ELEMENTARY GEOMETRY 255 

strictly deductive proof. Personally I would use it very 
much more for teaching even advanced geometry; I always 
did in my teaching days. For elementary work at all events, 
it is a singularly useful weapon. Though proof by means 
of it is difficult for beginners to set out, it produces con- 
viction in the beginner, a great gain. 

It will be observed that, for framing definitions, we have 
used the old device per genus et differentiam. This is probably 
the only safe method for beginners. From schoolboys we 
must be satisfied with something much less than perfection 
in their definitions. In particular, do not worry about " re- 
dundant " definitions. In the early stages they are inevitable; 
they are then almost to be encouraged. It is much better 
to let a young boy say that " a rectangle is a right-angled 
parallelogram " than " a rectangle is a parallelogram with 
a right angle ". A beginner naturally,., regards the latter 
with suspicion. It is doubtful wisdom ever to ask a boy to 
define a straight line or an angle. He has clear notions of 
these things already, and these notions he cannot express 
in language that is entirely satisfactory. If a boy says that 
" a straight line is the shortest distance between two points ", 
strictly the definition is unacceptable, because of the vague 
term " distance ". If he adds as tested by a stretched string, 
we should feel that the idea in his mind was clear and distinct; 
and what more can we want from him? As for an angle, I 
have often asked boys for a definition, not because I expected 
a satisfactory one, but in order to show them that, whatever 
definition they put forward, it was open to criticism. Who 
has ever defined either a straight line or an angle satisfactorily? 
Again: ordinarily we distinguish between a circle and its 
circumference, and a useful distinction it is. And yet we 
all talk about drawing a circle to pass through three points. 
However, matters of this kind are not for beginners but for 
the Sixth Form, which is the proper place for a final polishing 
up of all such things. 



256 CRAFTSMANSHIP IN MATHEMATICS 

Working Tools for Future Deductive Treatment 

These consist of the familiar propositions concerning: 

1. Angles at a point. 

2. Parallels. 

3. Congruency. 

4. Pythagoras.^ 

5. Circles; such properties as can be established 

from considerations of symmetry. 

The formal proof of Pythagoras is easily mastered in the 
Fourth Form, but proofs of the other theorems may wait 
until the Sixth. Meanwhile all the propositions must be 
thoroughly known as geometrical facts, facts which can readily 
be used and referred to in all subsequent work. Although 
formal proofs are beyond beginners, the probable truth of 
the propositions must be substantiated in some way. Justi- 
fication is always possible at this stage, though rigorous proof 
is not. Most of the more recent textbooks provide " practical " 
proofs of a kind which to the beginner really do seem to 
justify the claims made by the theorems. Here, little need 
be said about such proofs. 

First considerations of angles at a point naturally arise 
when the nature of an angle itself is being discussed. Acute, 
obtuse, adjacent, reflex, complementary, supplementary, 
and vertically opposite angles may all be brought into an 
early lesson, provided that the rotational idea of the angle 
is clearly demonstrated. Angles up to 360 should be con- 
sidered from the first. 

Here is a first lesson on parallels and transversals. 

Parallel Lines and Transversals 

76. You have already learnt that the blue lines on the 
pages of your exercise books are parallel, that is, they run 
in the same direction and are always the same distance 
apart. When we speak of " distance apart " we mean the 



ELEMENTARY GEOMETRY 257 

shortest distance, and that distance is represented by a 
perpendicular from one line to the other. But can we be quite 
sure that a line which is perpendicular to one of the parallel 
lines is also perpendicular to the other? 

77. A line that is drawn across two or more other lines 
is called a transversal (trans means across). Draw a trans- 
versal PQ across the parallel lines 

AB and CD, cutting AB in M and p \ 

CD in N. A 

Imagine a man to walk along AB 
and, on reaching M, to turn to the c 
right and walk along MN. He has 
turned through the angle a x , for Fig. 86 

he was first walking towards B, 

and is now walking towards Q. On reaching N, let him 
turn to the left, and walk along ND. He has now turned 
through the angle a 2 , for he was walking towards Q and 
is now walking towards D. But now that he is walking 
along ND he is walking in the same direction as when 
he was walking along MB. Hence the angle he turned 
through on reaching N is equal to the angle he turned through 
on reaching M, that is, the angle a 2 is equal to the angle a v 

We might have expected this, for the two angles a x and 
a 2 look alike. They are called corresponding angles. 

78. When a transversal is drawn across two 
parallel lines, the corresponding angles are equal. 
(L.) Hence, 

79. A transversal which is perpendicular to one 
of two parallel lines is also perpendicular to the other. 
(L.) Conversely, 

80. If two lines are both perpendicular to a 
transversal, they are parallel to 

each other. (L.) A 

81. Just as we showed that the 
corresponding angles c^ and a 2 are c 
equal, so we may show that the 
corresponding angles fa and fa are Fig . 87 

(E291) 18 



258 CRAFTSMANSHIP IN MATHEMATICS 

equal; also a 3 and a 4 ; also j8 3 and /? 4 . (Fig. 87.) But a 2 and a 4 
are also equal, because they are vertically opposite angles. 

Since a t = a 2 , ( 78) 

and since a 2 = a 4 , 
therefore a A = a 4 . 

The angles aj and a 4 are on opposite sides of the transversal, 
and are called alternate angles. 

Similarly it can be shown that the alternate angles fi 3 
and /9 2 are equal. 

82. When a transversal is drawn across two 
parallel lines, the alternate angles are equal. (L.) 

83. Observe that, in the eight marked angles of the 
last figure, there are four pairs of opposite angles, four pairs 
of corresponding angles, two pairs of alternate angles, every 
pair being equal. The four a's are equal, and the four fi's 
are equal. 

The four angles between the parallel lines are called 
interior angles. 

The four angles outside the parallel lines are called 
exterior angles. 

The following is very important: 

a i + Pa a straight angle, 
two right angles. 

But p 3 - p 2 , ( 82) 

therefore ^1+^2 two right angles. 

Similarly we may show that a 4 + /? 3 = two right angles. 

84. When a transversal is drawn across two 
parallel lines, the two interior angles on the same 
side of it are together equal to two right angles, that 
is, they are supplementary. (L.) 

Considerations of congruency are best led up to by actual 
practical work on the construction of triangles from given 
data. One lesson is enough for the boys to discover that a 
triangle can be described if 



ELEMENTARY GEOMETRY 



259 



(1) the 3 sides are given, 

(2) 2 sides and the included angle are given, 

(3) 1 side and 2 angles are given; 

md that therefore two triangles are congruent if there is 
:orrespondence and equality of 

(1) 3 sides, 

(2) 2 sides and the included angle, 

(3) 1 side and 2 angles. 

Further than this with beginners it is unnecessary to go. 

As for Pythagoras, it is enough to give beginners one or 
:wo of the many well-known dissection figures. 

Here T = W f V. 





Fi. 88 

Here the big square is cut up to form the two little 
squares, 

P = Q = R = s = P' =. Q' - R' - S', 
M = M'. 

The fact must be emphasized that in 
:hese early stages any attempt at 
formal proof is out of place. Never- 
:heless adequate reasons may be 
xnmd, and should be provided, in 
mpport of all statements made con- 
cerning these fundamental proposi- 
ions. 

Here is a lesson on the centre of the circle as a centre 
>f symmetry. 




260 



CRAFTSMANSHIP IN MATHEMATICS 



85. The centre of the circle as a centre of symmetry. 

Fold a circle on a diameter AB as an axis of symmetry, 
prick through the two halves at C, open out and call the 
corresponding points Q and C 2 , join C a and C 2 and join 
each to the centre O, thus forming the isos. A C 1 OC 2 . 
The arc of the sector AQOCg and the arc of the segment 
ACjCg are the same. 

Take a piece of celluloid (a piece of tracing-paper will 
do, if you use it carefully), pin it down over the circle (fig. 
90, i) by means of a pin thrust through it and through the 





Fig. 90 



centre O, and trace on the celluloid the radii OCj, OC 2 , 
the chord QCg, and the arc C 1 AC 2 \ now rotate the celluloid 
round the pin (fig. 90, ii). 

Since the traced sector and segment on the celluloid 
preserve their shape and size while rotating, all lengths 
and angles remain unchanged, and the arc C^ACg is seen 
always to fit exactly on the circumference below it. We say 
that the rotating sector and segment are symmetrical with 
respect to the centre of the circle, because of this exact 
fitting during the whole of a rotation. Thus the length of 
the arc, the length of the chords, the angle between the 
radii, all remain constant. We see all this plainly in fig. ii, 
where the rotating sector and segment are shown in two 
positions. Hence: 

86. Equal chords in a circle are equidistant from 
the centre. (L.) 



ELEMENTARY GEOMETRY 261 

87. Equal chords in a circle are subtended by equal 
arcs. (L.) 

88. Equal angles at the centre of a circle are sub- 
tended by equal chords and by equal arcs. (L.) 

89. If two chords of a circle are equal, they cut 
off equal segments. (L.) 

90. All these things ( 86-89) which apply to one 
circle also apply to equal circles, since equal circles will fit 
together exactly. 

Early Deductive Treatment 

Do not expect any rigorous logic from beginners. We 
suggest a lesson easily within the comprehension of young 
boys at the end of their First Year. Note: (1) the proofs 
though of the simplest kind are enough to convince young 
boys; (2) there is a logical grouping of the different kinds 
of parallelograms; (3) the gradual extension of the properties, 
as the variety within the species becomes more particularized, 
is brought out. This gradual extension of properties should 
always be borne in mind in the teaching of geometry. 

Quadrilaterals as Parallelograms 

91. A quadrilateral is a plane rectilineal figure 
with four sides. (L.) There are different kinds of 
quadrilaterals. We will begin 

with the parallelogram. / / / / / 

92. Draw a few parallel / / / / / 

transversals across the parallel / / / / / 

lines of your notebook. You / / / / / 

see a number of four-sided Fig. 91 

figures with their opposite sides 

parallel. These are parallelograms. (Gram means line.) 

How can we define a parallelogram? First put it into 
its class. ( 73.) 

A parallelogram is a quadrilateral . . . 



262 



CRAFTSMANSHIP IN MATHEMATICS 



What special property distinguishes a parallelogram? Its 
opposite sides are parallel. Hence the definition: 

A parallelogram is a quadrilateral with its opposite 
sides parallel. (L.) Now let us discover the other properties 
of a parallelogram. 

93. Any side of a parallelogram may be regarded as a 
transversal across two parallel lines. Let ABCD be a parallelo- 
gram, with the four angles 
marked as shown. 





Fig. 93 



a + p = 2 rt. Zs, 

p 4. y = 2 rt. Zs; 
/. a -f (3 = p + y. 

/. a - y. 
Similarly, p = 8. 



(84) 
(84) 



Thus, the opposite angles of a parallelogram are equal. 




Fig. 93 



94. Join two opposite vertices by a line. Such a line is 
called a diagonal. This diagonal (AC) is a new transversal 

to both pairs of parallel lines. 
Mark the two pairs of equal 
alternate angles, a l9 2 , ft, ft, 
and the pair of equal opposite 
angles, y 1? y 2 . In the two Z.s 
ABC, ADC, the diagonal forms 

a side belonging to both, and the three angles of the one 
are equal, respectively, to the three angles of the other. 
Hence the two As are congruent. 

.'. AB = CD and AD = BC. 

Thus, the opposite sides of a parallelogram are 
equal. (L.) 

95. Draw the two diagonals, intersecting at E; and 
consider the two triangles AEB, CED. In these As, the 
three /.s of the one are equal to the three /.s of the other 
( 82), and AB = CD ( 94). Hence the As are congruent. 

/. AE = CE and BE = DE. 



ELEMENTARY GEOMETRY 



263 



Thus, the diagonals of a 
parallelogram bisect each 
other. (L.) (Of course we 
might have used the other As 
AED and EEC.) 

96. We may now collect 
up the various properties of a 
parallelogram: 




Fig. 94 



In any Parallelogram, 

(1) The opposite sides are parallel, (known from the 



(2) The opposite angles are equal. 

(3) The opposite sides are equal. 



definition) 
(proved) 
(proved) 



(4) The diagonals bisect each other, (proved) 

97. Imagine a 17 m to be jointed at the four angles, and 
let it move from position A to position C. It remains a 



Fig. 95 

ZZ7m all the time, and therefore keeps all its properties. But 
on its journey to C it passes through B, where the angles 
are right angles. A parallelogram with right angles is called 
a rectangle. A RECTANGLE is a right-angled parallelo- 
gram. (L.) Since a rectangle is a ZI7m, it has all the 
properties of a I7m ( 96); and it has certain additional 
properties: 

98. (i) The angles of a rectangle are right angles. 
(This follows from the definition.) 

99. (ii) Draw the two diagonals of 
the rectangle ABCD, and examine the 
two As ABC and DCB (which, it will 
be seen, partly overlap and have a 

common base). The two sides AB, BC are equal to the 




Fig. 96 



264 CRAFTSMANSHIP IN MATHEMATICS 

two sides DC, CB ( 96, 3); and the included angles ABC 
and DCB are equal, both being right angles. Hence the 
As are congruent. Therefore AC is equal to BD. Thus, 
the diagonals of a rectangle are equal. (L.) 

100. Suppose a rectangle gets shorter and shorter until 
its length and breadth are equal. It remains a rectangle 



Fig. 97 

all the time and therefore keeps all its properties. When 
the length exceeds the breadth, as in A and B, the rectangle 
is called an oblong; when its length and breadth are equal, 
it is called a square. 

101. An OBLONG is a rectangle with its length 
exceeding its breadth. (L.) 

102. A SQUARE is a rectangle with all four sides 
equal. (L.) Since a square is a rectangle, it has all the 
properties of a rectangle ( 97-99); and it has certain 
additional properties: 

103. (i) All four sides of a square are 
equal. (This follows from the definition.) 

104. (ii) Draw the two diagonals of the 
square ABCD, intersecting in E, and examine 

Fig. 9 8 the two As EEC and DEC. 

BE = DE, ( 95, 97) 

BC - DC, ( 103) 

EC is common to both As; 
/. ABEC = A DEC; 
.'. ZBEC = /.DEC; 

.*. each Z = J st. Z or 1 rt. Z. 

Similarly we may show that each of the other two Zs at 
E are rt. Zs. Hence, the diagonals of a square bisect 
each other at rt. Zs. (L.) 

105. We still require names for the two non-rectangular 
parallelograms. The non-rectangular parallelogram with all 
four sides equal is called a rhombus (fig. 99, i). The non- 




ELEMENTARY GEOMETRY 



265 



Fig. 99 



rectangular parallelogram with only its opposite sides equal 
is called a rhomboid (ii). The rhomboid is the parallelogram 

we began with ( 93-96). It 

is the most general form of / / / / 

parallelogram. If we made all i 

its sides equal, or if we made 

all its angles right angles, we should make it a particular 

kind of parallelogram. 

106. Just as an oblong may be reduced in length and 
made a square, so a rhomboid may be reduced in length 
and made a rhombus. 

Just as a square has all the properties of an oblong and 
certain additional properties, so a rhombus has all the pro- 
perties of a rhomboid and certain additional properties. 

107. All four sides of a rhombus are equal. (This 
follows from the definition.) 

108. The diagonals of a rhombus are the perpen- 
dicular bisectors of each other. (L.) This property 
may be discovered in this way. The word 

rhombus really means a spinning-top. If 

we stand it on an angle, it looks something 

like a spinning-top. We see at once that 

either diagonal is an axis of symmetry, B 

and D being corresponding points about 

the axis AC, and A and C being corresponding points about 

the axis BD. Thus each diagonal is the perpendicular 

bisector of the other. 

But this also applies to a square. What is the difference 
between a rhombus and a square? 






Fig. 101 



109. If we lengthen (or shorten) equally the two halves 
of one of the diagonals, AB, of a square, we stretch out (or 



266 CRAFTSMANSHIP IN MATHEMATICS 

contract) the square into a rhombus. The diagonals of a 
square are equal; those of a rhombus are unequal. 
A rhombus differs from a square in these ways: 

1. Its angles are not right angles. 

2. Its diagonals are not equal. 

A rhombus resembles a square in these: 

1. The four sides are equal. 

2. Each diagonal is the perpendicular bisector of the 

other. 

3. Each diagonal is an axis of symmetry. 

110. We may classify the four kinds of parallelograms in 
this way: 

Parallelograms 



Rectangles Non- rectangles 



Square Oblong Rhombus Rhomboid 

(all sides eq.) (only opp. sides eq.) (all sides eq.) (only opp. sides eq.) 

How easy it is to make up definitions from this scheme: 

111. A SQUARE is a rectangular parallelogram 
with all four sides equal ( 102). 

112. An OBLONG is a rectangular parallelogram 
with only its opposite sides equal ( 101). 

113. A RHOMBUS is a non -rectangular parallelo- 
gram with all four sides equal. (L.) 

114. A RHOMBOID is a non -rectangular parallelo- 
gram with only its opposite sides equal. (L.) 

115. We might classify parallelograms according to their 
axes of symmetry. A square has four axes of symmetry, viz. 
two diagonals and two medians (a median is the line joining 
the middle points of opposite sides); an oblong has two, 



ELEMENTARY GEOMETRY 267 

viz. the two medians; a rhombus has two, viz. the two 
diagonals; a rhomboid has none. But in definitions we 
do not usually refer to symmetry; symmetry is useful mainly 
for discovering properties. 

Remember that the two halves of a figure folded on an 
axis of symmetry will fit together exactly. Remember, too, 
that a figure can always be imagined to spin on an axis of 
symmetry. 

116. More about definitions. We have defined a 
square as a rectangle with all four sides equal. There- 
fore, 

(i) Since it is a rectangle, it has right angles and is a 
parallelogram ( 97). 

(ii) As it is a parallelogram, its opposite sides are 
parallel ( 92). Thus, our definition of a square tells 
us three things: 

1. The four sides are equal, 

2. The four angles are right angles, 

3. The opposite sides are parallel. 

But the definition tells us nothing at all about the diagonals. 
Properties of the diagonals must be discovered either by 
congruence ( 104) or by symmetry. 

117. We might define a square as a quadrilateral with 
four equal sides and four right angles. But all that this defini- 
tion tells us is that: 

1. The four sides are equal, 

2. The four angles are right angles. 

It is quite a good definition, but it does not tell us that the 
square is a parallelogram, and therefore it does not tell 
us that the opposite sides are parallel. Hence this property 
is one we should have to find out (perhaps by congruence) if 
we used the new definition. Let us decide not to use it. 

118. We can now classify the properties of parallelo- 
grams: 



268 



CRAFTSMANSHIP IN MATHEMATICS 





Rhomboid. 


Oblong. 


Square. 


Rhombus. 




3 


a 


D 


n 


1. Opp. sides ||. 


X 


X 


X 


X 


2. Opp. sides eq. 


X 


X 


X 


X 


3. Opp. Zs. eq. 


X 


X 


X 


X 


4. Diags. bisect each other. 


X 


X 


X 


X 


5. All four Zs. rt. Zs. 




X 


X 




6. Diags. eq. 




X 


X 




7. All four sides, eq. 






X 


X 


8. Diags. at rt. Zs. 






X 


X 



Observe that the rhomboid is the most general of the parallelo- 
grams, and has fewest properties; and that the square is the 
most special of the parallelograms, and has most properties. 
No other parallelogram has all the properties of the square. 

This lesson may usefully be followed up by the considera- 
tion of quadrilaterals that are not parallelograms. 



Proportion and Similarity 

A knowledge of proportion and similarity is so fruitful 
throughout the whole range of the study of geometry that 
the subject should be introduced at an early stage, though 
naturally incommensurables are then ignored entirely. We 
append a lesson suitable for the second year of the geometry 
course. Note the little device for constructing a triangle 
with sides simply commensurable. The proofs given are 
rigorous enough at this early stage. The important thing is 
to provide learners with a serviceable weapon rough and 
unpolished, for the moment, it is true; but that is of no 
consequence. 

(The nature of a ratio, of cross-multiplication, &c., has 
already been referred to in the chapters on arithmetic and 
algebra, but the three subjects should be brought into line 
when a principle common to them all is under consideration.) 




ELEMENTARY GEOMETRY 269 

119. Take a piece of paper ruled in J" squares, and on 
it draw this triangle: the base AB of the A is to be on one 
of the ruled horizontal lines 4" or 5" down the paper, and 
the vertex in a parallel line 3" 

above, i.e. in the twelfth parallel 

line above. Fix the point A 

towards the left-hand end of the 

line selected for the base, and with 

a radius of 3-6" draw a circle to 

cut the top line in C. With C Fig. 102 

as centre, and with a radius of 

4-5", draw a circle to cut the base line in B. Join AB (its 

length does not matter) and so complete the A ABC. 

Since AC = 3-6", it can be divided into twelve equal 
parts of -3" each, and each division will fall on one of the 
ruled horizontal lines. Since CB ^ - 4i", it can be divided 
into twelve parts of f " each, and again each division will fall 
on one of the ruled horizontal lines. But the ruled lines are 
all parallel to each other. We therefore seem to have the 
following result: 

120. If the two sides of a triangle are divided into 
the same number of equal parts, and the corresponding 
points of division in the two sides are joined, all the 
joining lines are parallel to the base. (L.) 

It has been found that this result is always true, no matter 
how it is tested. But the real proof is too difficult for you 
to understand at present. The following particular case is 
often useful: 

121. If two sides of a triangle are bisected, the line 
joining the points of division is parallel to the base. 
(L.) 

122. We may now learn that if 
a line is drawn parallel to one side 
of a A , it cuts the other side pro- 
portionally. Consider, for instance, 
the fifth parallel DE, from the top 

in the figure to 119. CD is - x \ of Flg> I03 




270 CRAFTSMANSHIP IN MATHEMATICS 

CA, and DA is -& of CA; CE is -& of CB, and EB is -/ 2 - of 
CB. 

CD 5 , CE 5 

nnn 

" DA~7 EB~7' 

CD ^ CE 

" DA " EB' 

So with any other parallel. Or a part of a side may be compared 
with the whole. For instance, 

CD ^ CE 

CA ~ CB' 

for each is equal to the fraction - L ! V 

We can imagine the A CDE to be a small A fitting over 
the top of the larger A CAB (fig. 104, i), and CD being made 
to slide down CA so that the small A CDE occupies the 



\L _\r 




Fig. 104 



position C'D'E', D taking the place of A (fig. 104, ii). Just 
as the three /.s of the small A are respectively equal to the 
three Ls of the large A in fig. 104, i (see 78), so they 
must be in the second, since corresponding angles are equal. 
Hence C'E' is || CB, and therefore C'E' cuts the two sides 
D'C and D'B proportionally; and just as C'D' is -{\ of CD', 
so D'E' must be ^ of D'B, 

D'C' D'E' D'C' D'E' 

or = . or = . 

C'C E'B' D'C D'B 

Similarly by making the little A slide down to the other 
corner (fig. 104, i to iii), so that E takes the place of B, C"D" 
is || to CA, and therefore 

E // C " E"D" E"C" E"D" 

- . Q| 

C"C D"A' E"C E"A' 

Hence: 





ELEMENTARY GEOMETRY 271 

123. If in a triangle a line is drawn parallel to 
any side, it cuts the other sides proportionally. (L.) 
(This is always true, but the real proof is too difficult for 
you to understand at present.) 

124. We might detach the small ACDE from the large 
one CAB, and place the two side by side. They look alike 
They are alike. They are similar. c 

Although the sides of the two As 
differ so much in length, the three 

Z.s of the one are respectively equal to A / \ B 

the three /.s of the other. (Why?) b lg . I05 

In other words, the two similar As are 

equiangular. And we know already that the corresponding 
sides are proportional. This we should expect in similar 
figures of any kind. In a photograph of yourself, for 
instance, you would expect the " proportions " of your body 
to be accurately preserved. If the ratio of the lengths of 
your outstretched forearm and upper arm is f , you would 
expect that ratio to be preserved in the photograph (or the 
photographer would probably hear about it!). 

125. If, then, ABC and DEF are two similar As, the 
corresponding sides are proportional. But note that we 
may express the ratios in two 

different ways: (1) two sides of 
one A as a ratio equal to the 
ratio of the corr. two sides of the 
other A ; (2) one side of one A p lfg . I0 6 

and the corr. side of the other 

A as a ratio equal to the ratio of any second side of the 
first A and the corr. side of the second A . Consider, for 
instance, the two sides AB, BC in the A ABC, and the two 
corr. sides DE, EF in the A DEF. We may say, 

AB DE AB BC 

BC'EF' r DE = EF- 

The two proportional statements are really the same thing 




272 CRAFTSMANSHIP IN MATHEMATICS 

since we obtain the same product from the cross-multi- 
plication of either: 

AB . EF = BC . DE. (The full stop is used instead of X .) 

It is often an advantage to interchange one form for the 
other; really we interchange the second term of the first 
ratio and the first term of the second. 
We have learnt that: 

126. SIMILAR TRIANGLES are equiangular, and 
their corresponding sides are proportional. (L.) 

127. When expressing ratios between two sides of each 
of two similar As, be careful to select corresponding 






sides, i.e. sides taken in the same order round corre- 
sponding angles. In these two pairs of As, the corre- 
sponding Zs are marked with the same Greek letters. From 
the first pair we may equate ratios thus, six equations in all: 

AB = DE. AB _ DE. AC _ OF. AB _ BC. AB _ AC. AC _ BC 
BC EF ; AC DF ; CB FE J DE EF ; DE DF ; DF EF* 

From the second pair, we may do exactly the same thing: 

=Q5. MP^QS. MN^NP. M^MP. MP__PN 
QS ; PN SR ; QR RS ; QR QS ; QS SR* 

Yet there appears to be a difference. That is because in 
the second pair the As are right- and left-handed. If you 
have any doubt, turn one of the pair over, through 180, 
as you would turn over a page of a book. Then the pair 
will look alike. But if you mark the corresponding angles 
correctly, you ought to have no difficulty. 



ELEMENTARY GEOMETRY 



273 



The sides of the triangles may usefully be named by 
means of single small letters; then the writing of the ratios 

is simplified; e.g. - instead of .. The small letter selected 
a BC 

for a side is always the same as the capital letter naming the 
opposite angle. 

128. Sometimes each of a pair of similar As is similarly 
divided by a perpendicular from a vertex to the opposite 
side. The resulting pair of A 

As in the one case are evi- 
dently similar, respectively, 
to the resulting pair in the 

other case, for they are 

. ~~ u ~ " 

equiangular, i.e. As ABG iv ]g . 10 s 

and DEH are similar, and 

As ACG and DFH arc similar (check, by sum of angles). 
We may reason in this way: 




In the As ABG, DEH, since 



AG_ = AB 
DH DE 

4, 4, 

AB 



and in the As ABC, DEF, since = 

I 
. AG _ BC 

" DH EF* 



An exchange of ratios may often usefully be made in this 
way. From this particular exchange we learn that the altitudes 
AG, DH are proportional to the bases BC, EF. Hence: 

129. When similar triangles are divided by perpen- 
diculars drawn from corresponding vertices to opposite 
sides, an exchange of ratios may often be usefully 
made. (L.) 

130. In similar triangles, the altitudes are pro- 
portional to the bases. (L.) 

131. We know that when two ratios are equated to form 
a proportion, they may be cleared of fractions by cross- 

(E291) 19 



274 



CRAFTSMANSHIP IN MATHEMATICS 



multiplication. For instance, in the two similar As ABC, 
DEF, we know that 

=- ; (an equation consisting of 2 ratios) 

BC EF 

/. AB . EF = DE . BC. (an equation consisting of 2 products) 

What does this mean? AB, BC, DE, EF, all represent lines 
of a particular length; a length multiplied by a length gives 
an area. Thus each of the two products AB . EF and DE . BC 
represents a rectangle. 




c e 




6-25 




Fig. 109 



Note that we begin with ratios, i.e. with quotients, 
representing a length divided by a length. After cross- 
multiplying, we have products representing areas, or a 
length multiplied by a length. (The measured lengths are 
shown to scale. Check the numerical ratios and the products. 

6-25 3'75 
For instance, are and equal? and are 10 X 3-75 

and 6X6- 25 equal?) 10 6 

132. By cross -multiplication, two equated ratios 
of lengths give two equated rectangular areas. 

(" Equated " means expressed as an equation.) 

133. The last result is useful in all 
sorts of ways. For instance, AB and 
CD are two chords of a circle, inter- 
secting at O. Join AC and DB, and 
we have two similar As, the As being 
equiangular (angles in the same segment; 
see 126). Taking ratios (see 127) 
we have, 




Fig. no 



OA 

oc : 

rect. OA . OB 



op 

OB ; 

rect. OC . OD. 



ELEMENTARY GEOMETRY 



275 



Hence, if two chords intersect in a circle, the rectangle 
contained by the two segments of the one is equal to 
the rectangle contained by the two segments of the 
other. (L.) (The term " segment " here applies to the 
parts of the chords.) 




Fig. in 

The proportional division of lines and the construction of 
similar figures should follow at a slightly later stage. Be sure 
that the boys master the principle exemplified in these three 
figures: for the construction of the similar pentagon the 
position of. the point O is quite immaterial. 

The centre of similarity problems are readily followed by 
those on centre of similitude. Insist on the point that any 
two circles may be regarded as similar figures, since, like 
rectilineal similar figures, they may be looked upon as the 
same figure drawn to different scales. 



Circles and Polygons 

The ordinary properties of the circle give little trouble 
angles in a segment, the cyclic quadrilateral, tangents, 
alternate segment property, circles in contact, and inter- 
secting circles. Do not forget to group properties around a 



276 



CRAFTSMANSHIP IN MATHEMATICS 



common principle; e.g. (i) the tangent to a circle, (ii) the 
external common tangent to two circles, (iii) the transverse 
common tangent to two circles, should be taken in that order, 
and be made to follow on the key proposition that the angle 
in a semicircle is a right angle. All these propositions on the 
circle being quite simple, formal proofs should now be 
consistently exacted. 

Regular polygons, too, need give little trouble. Their 
angle properties are interesting, easy to understand, and 
always appeal to a boy. The pentagon excepted (see the 
next section), they are not much 
wanted. The hexagon and octagon 
involve the simplest geometry, easy 
work for beginners. The decagon is 
easily constructed from the pentagon, 
and the dodecagon from the hexagon. 
The heptagon and nonagon are hardly 
ever used; the latter is easily constructed 
from its angle properties; the former 
is riot, inasmuch as its angle properties 
involve fractions of a degree and hence 
some sort of approximation method is 
required for its construction. The 

best is probably the following, especially as it is common to 
all polygons. 

On one side of a straight line draw a semicircle and on 
the other side an equilateral triangle. If the line be divided 
into x equal parts, and lines be drawn from the apex of the 
triangle through the points of division, to meet the semi- 
circle, the semicircle is divided into the same number of 
parts as the line. This is not susceptible of proof, simply 
because it is not mathematically true, but the approximation 
is so near that the most careful measurement usually fails 
to detect an error. Architects generally use it. Evidently 
by drawing radii from the points of division of the semicircle, 
we divide 180 into x equal parts. 

The conversion of polygons (regular and irregular) into 




ELEMENTARY GEOMETRY 277 

triangles, triangles into rectangles, and rectangles into 
squares, which is often wanted, is simple straightforward 
work, though some little practice in manipulating the figures 
is necessary. To the beginner, a polygon with one or more 
re-entrant angles is puzzling. 

Golden Section and the Pentagon 

We append the following lesson as an example of linking 
up different Euclidean propositions (II, 11; IV, 10, 11) and 
of utilizing algebra in solving geometrical problems. 

134. To divide a line into two parts so that the 
rectangle contained by the whole and one part is equal 
to the square on the other part. 

This is sometimes stated: 

To divide a line in medial section. 
or, To divide a line in extreme and mean ratio. 
or, To divide a line in golden section. 

The problem is very easy to do and to understand if we 
can solve easy quadratic equations. It is the kind of problem 
in which algebra can help us 

much, A , x L(2-z) > B 

Let AB be the line to Fig II3 

be divided, and let it be, say, 

2" long. Suppose the point of division is P. 

Let AP be x inches long; then PB = (2 x) inches long. 
The line has to be divided so that AB . BP = AP 2 , 

i.e. 2(2 x) = x 2 . 
We must now solve this equation, and find the value of x 

X * = 2(2 - *); 
.'. x 2 + 2x = 4. 
/. x 2 + 2x + I = 5. 

.*. x + I = + V5. (We may neglect the minus sign.) 
/. X = V5 - 1, 



278 



CRAFTSMANSHIP IN MATHEMATICS 



i.e. AP = (\/5 1) inches. Can we measure off this length 
and so find P? Yes, by the theorem of Pythagoras. We do 
it in this way: 

Erect a JL BC at B, 1" long, and join CA. 

AC 2 - (AB 2 + BC 2 ) = (2 2 + I 2 ) = 5; 
/. AC = V5, 
i.e. AC is V5 inches long (fig. 113a, i). 





Fig. 1130 

But we require a line (\/5 1) inches long. 
Since CB I", with centre C and radius CB, cut CA in 
D (fig. 1130, ii); CD = I". 

Thus AD - (V5 - 1) inches. 

But we require a part of AB equal to (\/5 1) inches. 
Hence, with centre A and radius AD, cut AB in P; 
AP = (<v/5 1) inches. 

Thus P is the point required. 

The length of PB is evidently 2 (<v/5 1) in., i.e. 
(3 V5) inches. 

If we have done " surds " in algebra, we can show 
that the result is correct: AB . BP has to be equal to 
AP 2 . Now AB . BP - 2(3 - ^5) = 6 - 2 V5; and AP 2 - 
(V5 - I) 2 = 6 - 2-v/5, as before. 

Here is Euclid's figure. He does not 
cut off a piece from CA; he makes CD 
equal to CA, so that CD = <v/5, and BD 
== \/5 ! Then he makes BP equal to 
BD = \/5 1, so that P is found as before, 
except that PB is now the longer instead 
Fig. ii 4 of the shorter section. The shaded parts 

of the figure show the rectangle AB . AP 
2(3 V 5 ); and the square on PB, (^/5 I) 2 . 




ELEMENTARY GEOMETRY 



279 



Now examine a regular pentagon and its 5 contained dia- 
gonals. Give the boys a few hints (such as the following) 
and then leave them to construct the pentagon themselves. 



(1) Angles. 04 = a 2 = a 3 ; hence it is clear that the 
15 angles at the 5 vertices of the pen- 
tagon are all equal, and that each = 

36. 

(2) Lines. Each diagonal is divided 
by 2 others into 3 parts. Is there 
any relation between the parts? e.g. 
does CF bear any relation to FA or to 
the side CD? 

Draw the triangle ACD and the 
line FD separately, and write in all 

the angles. Evidently AFD and DFC are isosceles triangles. 
. . AF = FD, CD = FD, / . AF = CD. Hence if we put 
a circle round the triangle AFD, CD is a tangent (relation 




Fig. 115 





Fig. 116 



= ?); also CA is a secant; / . CF . CA = CD 2 ; / . CF . CA 
= FA 2 , i.e. CA is divided at F in golden section. 

To construct a pentagon, therefore, we begin by drawing 
any line AC, and dividing it in golden section in F. With 
A as centre and AC as radius, we draw a circle (not shown) 
and draw in it the chord CD equal to FA, and then join 
AD. This gives us the triangle ACD, round which we 
circumscribe a circle and so obtain part of fig. 115; to obtain 
the points B and E we bisect the angles ADC, ACD. 



280 CRAFTSMANSHIP IN MATHEMATICS 

Teach the boys one or two special ways of drawing the 
pentagon; e.g. let them tie into a simple knot a strip of paper 




of uniform width. It is a useful exercise to make them prove 
that the figure produced really is a pentagon. 

The Principle of Continuity 

This is an ambiguous term, for in each of several branches 
of knowledge it is given a special significance. Even in the 
single subject mathematics, it is used in different senses. 
One standard textbook of geometry states: " The principle 
of continuity, the vital principle of modern geometry, asserts 
that if from the nature of a particular problem we should 
expect a certain number of solutions, and if in any particular 
case we find this number of solutions, then there will be 
the same number of solutions in all cases, although some of 
the solutions may be imaginary. For instance, a straight 
line can be drawn to cut a circle in two points; hence we 
state that every straight line will cut a circle in two points, 
although these may be imaginary or may coincide. Similarly 
we may say that two tangents may be drawn from any point 
to a circle, but they may be imaginary or coincident/' * 

But in geometry the term " continuity " has come to be 
used more loosely than that. It is used to indicate generality, 
a generalizing of some fundamental principle, or the grouping 
of a number of allied instances around some central principle. 
We give a few instances of different kinds, from which the 

* Lachlan, Modern Pure Geometry. 



ELEMENTARY GEOMETRY 



281 



reader will see more clearly what is meant. As regards the 
teaching of geometry, the principle is one of the very greatest 
importance. 

1. The particularizing of a general figure and the extension 
of properties. We have already given an instance of this in 
the lesson on parallelograms. 

2. Varying the figure to include different cases. These 
three figures tell their own story. If the parts of such figures 




Fig. 118 



are similarly named, as a rule exactly the same words apply 
in all cases to the proof. What difference there is is generally 
a difference of mere sign. 

3. Generalizing a term to include its natural extensions, 




Fig. 119 



e.g. a chord as a secant, and a secant as a tangent. From the 
case of intersection O inside the circle, we pass to the case of 
intersection outside the circle, and then from the two secants to 
a secant and a tangent. The three cases may first be separately 



282 



CRAFTSMANSHIP IN MATHEMATICS 



taken and then generalized. If the lettering is consistent, 
the arguments are identical, though for the tangent-secant 
case we should generally argue rather differently. In all 
three cases we have two similar triangles, OAC and OBD, 
and OA/OC = OD/OB. 

Another general chord-secant-tangent property is seen in 
the following four figures, showing the measure of an angle 




Fig. 1 20 

inscribed in a circle by reference to the intercepted arcs; 
again the argument may be made perfectly general. 

4. The extensions of Pythagoras form another series. It 
is the general custom nowadays to give the boys Pythagoras 
towards the end of their first year, to serve as a useful working 
tool; to give a formal proof during the second year, and to 
take the extensions (Euclid II, 12, 13) a few months later 
still; most boys are then familiar with the results in the 
following form. 



AB 2 = AC 2 -1- CB 2 exactly. 




AB 2 = 
AC 8 + CB 2 + 2BC.CP 



AC 8 + CB - 2BC.CP 



where CP is the 
projection of 
CA on BC. 
(L.) 



ELEMENTARY GEOMETRY 



283 



But figures to illustrate the extensions are less often pro- 
vided. Here is a suggestion: 




Fig. 121 



Fig. (i) illustrates Pythagoras, In (ii) compare the squares 
on the new sides BY, YA with the squares on the old sides 
BC, CA. In (iii) compare the squares on the new sides 
BZ, ZA with the squares on the old sides BC, CA. The 
dissections are interesting, though they tend to puzzle slower 
boys. 

5. Summing the exterior angles of a polygon: " walking 
the polygon ". First consider an ordinary convex polygon. 
Mark in the angles systematically: " always turn to the 



284 CRAFTSMANSHIP IN MATHEMATICS 

left "; the angle to be worked is that between the old 
direction and the new. 



SUM op Ls. 2ir 



Fig. 1220 

Secondly, a polygon with one re-entrant angle: 





SuMOfU-4ir 



Fig. 1226 

Thirdly, a cross polygon; also with one re-entrant angle: 

c 



D SUM op Ls 




Fig. I22C 

The point in this example is to see how exactly the same 
principle is followed out: always turn to the left, always 
measure the exterior angle between the old direction (pro- 
duced) and the new. The result must always be a multiple 

Of 27T. 

6. Euclid, Book IL Given an algebraic basis, suitable 



ELEMENTARY GEOMETRY 285 

figures, and a rational grouping, props. 4 and 7, 5 and 6, 
9 and 10, can be taught in a single lesson. Never make the 
boys go through the Euclidean jargon; life is not long enough. 
7. The Sections of a Cone (for more advanced pupils). 
Let a plane perpendicular to the plane of the paper rotate 
round the point P, first cutting the cone ABC parallel to 
the base; then obliquely to cut the slant surface; then more 
obliquely, parallel to AC and cutting the base; then per- 
pendicularly to the base and cutting the base. Since the 
motion may be regarded as continuous, we should expect 
no sudden changes in the properties of the curves made by 
the rotating plane as it cuts the cone. Why 
should there be? The boys' knowledge of * 

geometry ought by this time to make them P / \ 
revolt against the idea of any fundamental 
differences in the properties of the curves. 
The curves may all be described as conies 
possessing certain common properties. In 




\ 



particular positions the curves have certain Fig. 123 

additional and special properties, but the 
common properties will remain. Let the boys understand 
that for convenience we study the curves separately first, 
and collectively later. But make them see at the outset 
that the circle is just a particular case of an ellipse, just 
as the ellipse is a case of the more general conic. The 
elliptic orbit of the earth, for example, is so very nearly a 
circle that a correct figure drawn on paper is virtually in- 
distinguishable from a circle. Astronomical figures are 
often purposely exaggerated. 

8. The Polyhedra (see Chapter XXXVIII). These form 
an even better illustration of the principle of continuity than 
those already cited. 

The principle applies, in fact, to the whole range of geo- 
metry. To deal with a proposition as an unrelated unit is, 
generally speaking, to offend almost every canon of geometrical 
teaching. 



286 



CRAFTSMANSHIP IN MATHEMATICS 



The Principle of Duality 

This is best exemplified by a few well-known pairs of 
theorems: 



1. If the sides of a triangle 
are equal, the opposite angles are 
equal. 

2. If two triangles have two 
sides and the included angle 
respectively equal, the triangles 
are congruent. 

3. If a quadrilateral be in- 
scribed in a circle, the sum of 
one pair of opposite angles is 
equal to the sum of the other pair. 

4. If a hexagon be inscribed in 
a circle, the three points of in- 
tersection of pairs of opposite 
sides are collinear. 



1. If two angles of a triangle 
are equal, the opposite sides are 
equal. 

2. If two triangles have two 
angles and the included side 
respectively equal, the triangles 
are congruent. 

3. If a quadrilateral be circum- 
scribed about a circle, the sum of 
one pair of opposite sides is equal 
to the sum of the other pair. 

4. If a hexagon be circum- 
scribed about a circle, the three 
diagonal lines connecting opposite 
angles are concurrent. 



Such pairs of propositions are said be dual or reciprocal. 

There is, in short, a remarkable analogy between de- 
scriptive propositions concerning figures regarded as as- 
semblages of points and those concerning corresponding 
figures regarded as assemblages of straight lines. Any two 
figures of which the points of one correspond to the lines 
of the other are said to be reciprocal figures. When a pro- 
position has been proved for any figure, a corresponding 
proposition for the reciprocal figure may be enunciated by 
merely interchanging the terms point and line; locus and 
envelope*, point of intersection of two lines and line of inter- 
section through two points] &c. The truth of the reciprocal or 
dual proposition may usually be inferred from what is called 
" the principle of duality ". 

The teacher should always be on the look-out for examples 
of this principle which gives boys so much insight into geo- 
metry. Numerous examples of concurrency and collinearity 
will occur to him at once. The principle is especially useful 



ELEMENTARY GEOMETRY 287 

in the treatment of more advanced work, for instance in the 
theory of perspective and in the theory of the complete 
quadrilateral (tetrastigms and tetragrams). 



CHAPTER XXI 

Solid Geometry 

Preliminary Work 

First notions of solid geometry will have been given in 
the Preparatory School. Even in the Kindergarten School 
the children are made acquainted with the shapes of common 
geometrical figures and solids. Lower Form arithmetic is 
closely linked up with practical mensuration, and quite young 
boys are made familiar with the methods of measuring up 
rectangular surfaces and solids. The practical mensuration 
associated with early measurements in physics forms another 
introduction to solid geometry. First notions of projection 
are given in early geography lessons; very young boys soon 
acquire facility in building up vertical cross-sections from 
contoured ordnance maps, and when projection is first formally 
taken up in the mathematical lessons, say in the Pythagoras 
extensions or in early trigonometry, the main idea is already 
familiar. All the way up the school, three-dimensional 
geometry in some form should be made to serve as a hand- 
maid to the plane geometry. Indeed, first notions of the 
geometry of the sphere are required at a very early stage in 
the teaching of geography, and if these notions are to be 
properly implanted the mathematical Staff should make 
themselves responsible, for not all geography teachers are 
mathematicians. 

Only a minority of boys acquire readiness in reading 
geometrical figures of three dimensions. With the majority, 



288 CRAFTSMANSHIP IN MATHEMATICS 

the training of the geometrical imagination is a slow business. 
For the clear visualization of the correct spatial relations in 
an elaborate three-dimensional figure, or for that matter even 
in a simple one, models of some kind are, in the earlier stages, 
essential. 

Supplies of useful little wooden models of the geometrical 
solids are often found in the physics laboratory, though why 
physics teachers so frequently relieve their mathematical 
colleagues of this particular work I have never been able to 
discover. If models in wood are not available, models may 
be readily cut from good yellow bar soap; the material is 
cleaner to handle than raw potato or clay or plasticine. By 
means of a roughly-cut model, the correct shape of a trans- 
verse section of a geometrical solid can be realized at once. 
Personally I prefer models made from " nets " of cartridge 
paper or thin cardboard; these are easy to make and are 
permanent, but the making consumes a good deal of time. 

Useful skeleton models are readily made from pieces of 
long knitting needles, sharpened at each end and thrust into 
small connecting corks. Two slabs of cork to represent 
the Horizontal and Vertical Planes, tacked to a pair of hinged 
boards, and a few pointed knitting needles, make excellent 
provision for the first lessons on orthographic projection. 

The natural sections of an orange, or the cut sections of 
a well-shaped apple, are useful when teaching the geometry 
of the sphere. 

The small varnished wooden models of the cylinder, 
sphere, and cone, of the same diameter and height, are useful 
for showing, by displacement of water in a measuring jar, 
that the volumes are 3:2:1. 

A slated sphere, mounted, should be part of the equipment 
of all mathematical teachers. 

Even such a simple device as two intersecting sheets of 
paper, each sheet being slit half-way across, to show the 
intersection of two planes at any angle, is often useful. 

But of course all these props should gradually be with- 
drawn, and the eye made to depend on two-dimensional 



SOLID GEOMETRY 289 

drawings. Still, it is always an advantage, even for the trained 
mathematician, to put a few shading lines into such drawings. 
They help the eye greatly. 

Stereographic photographs, or even hand-made stereo- 
grams, are also a great aid in teaching solid geometry. These 
are easily provided, and stereoscopes are cheap. Mr. E. M. 
Langley used them with great effect as far back as the nineties. 

Do not forget that even for plane geometry models may 
be useful. The pantograph is particularly useful when teaching 
similarity (see Carson and Smith's Geometry). When teaching 
loci, encourage the boys to make wooden or cardboard 
" linkages " to represent engineering motions and astro- 
nomical movements. The loci are then given a reality. 

The boys should also be encouraged to make " nets " 
of the commoner geometrical solids, in cartridge paper or 
cardboard. Boys of 11 or 12 learn to make these readily, 
and at that age time can be spared. I have known boys of 
10 make almost perfect paper models of the five regular 
polyhedra. 

In naming triangular pyramids, name the vertex first, 
then the three corners of the base, thus, A. BCD. Note that 
any corner of such a pyramid may be regarded as a vertex, 
the other three being the corners of the base (just as any 
corner of a A may be regarded as a vertex, and the other 
two corners as the ends of the base). 

A problem like the following is better understood if a 
prism is actually cut up, perhaps a wooden one made in the 
carpenter's shop; or one may be cut neatly from a bar of 
soap. It is easier to cut up the latter with a thin knife 
than to cut up the former with a saw. 

A suitable model for showing that a prism is equal to 
three times the volume of a pyramid on the same base and of 
the same height is a little troublesome to make. Inasmuch as 
it is particularly useful in the demonstration of that important 
principle, we give a few hints for constructing it. 

Fig. 124, i, represents the complete triangular prism, 
with bases ABC, DBF. From it, cut the pyramid E.ABC 

(E291) 20 



290 



CRAFTSMANSHIP IN MATHEMATICS 



by holding the knife (or saw) at E, and cutting through to 
AC. iii shows the pyramid cut off. 

Now we have to cut the remaining piece (ii) into two 
other pyramids. Cut from it the pyramid C.DEF. To do 




this, hold the knife again at E, and cut down to DC. v shows 
the pyramid cut off, its new face being shaded, iv shows 
the part left. It is a curious-looking wedge-shaped pyramid. 
We will name it E.DAC. 

We may show (since pyramids on equal bases and of the 
same vertical height have the same volume) that the three pyra- 
mids (iii, iv, v) are equal in volume. The bases of iii and v, 
E.ABC and C.DEF, are equal, since they are the bases of the 
prism; and the heights are equal, for FC EB, and these are 
two of the long edges of the prism. Again, if we name iv and 
v E.ACD and E.FDC, we see that the bases are equal, for 
they are the halves of DACF, one of the faces of the prism; 
and their vertical heights are equal, since the two have a 
common vertex E. Hence the volumes of all three pyramids 
are equal. 

It is a simple matter to make the " nets " (fig. 125) of 
the three pyramids, and fold them up to make models. The 
models may then be placed together to form the prism. Each 
net will, of course, consist of four triangles, the sides of all 
of which will be edges, or diagonals of the faces, of the prism. 

The formal mensuration of geometry of the pyramid, then 
of the cone, then of the cylinder, is interesting and valuable, 



SOLID GEOMETRY 



291 




Fig. 125 

and I do not find that it gives teachers much trouble, always 
provided that the necessary preliminary work from Euclid 
XI on lines and planes has been done well. 

The calculation of the areas of the surfaces of solids is 
also simple, including even the surface of the sphere, provided 
that suitable figures are drawn. 



Euclid XI 

All the essential propositions from Euclid, Book XI, are 
now included in the leading schoolbooks on geometry. 
Most boys find the reasoning easy enough, but many have 
great difficulty in understanding the figures, unless models 
are available to help visualization. 

It will suffice to touch upon Euclid XI, 4 and 6. 

XI, 4. If a straight line is perpendicular to each of two 
intersecting straight lines at their point of intersection, it is 
perpendicular to the plane 
containing them. 

The first of the follow- 
ing figures is Euclid's 
own, and to most boys 
it is incomprehensible. 
The second is that found Fig I26 

in many modern text- 
books. This is a case where a model is certainly desirable. 
Failing that, two figures should be drawn, from which the 




292 



CRAFTSMANSHIP IN MATHEMATICS 



different planes may easily be picked out. The following 
figures are suitable: in the first, the horizontal plane and the 
vertical planes are easily seen; in the second, the two oblique 
planes. If such figures are steadily gazed at, with one eye, 





Fig. 127 

through a very small hole in, say, a piece of cardboard, 
they quickly assume an appearance of three dimensions. 

XI, 6. If two straight lines are perpendicular to the same 
plane, they are parallel. 

The first figure is Euclid's (again a poor thing); the 
second is that commonly found in school textbooks. In 






Fig. 128 

this case, again, the planes want sorting out, to help visual- 
ization. Figure 129 is more suitable, with the horizontal 
plane shaded. The two perpendiculars AB, CD are shown 
by rather thicker lines. The two congruent triangles FDA, 
EDA in the oblique plane AFE are easily picked out; so 
are the two BDE, BDF in the horizontal plane. But it is so 
difficult to draw a figure that will show, to a beginner's eye, 



SOLID GEOMETRY 293 

the two congruent triangles ABE, ABF in their separate 
vertical planes, that either a wire model or a pair of stereo- 
grams are certainly desirable. 




Fig. 129 

It is unfortunate that so many boys experience difficulty 
in visualizing three-dimensional figures. But the fact has 
to be allowed for, and provision made accordingly. 

Do not press too far the argument that such aids as models 
should be withdrawn in order that the boys' imagination may 
be given opportunity to develop. The boys' developed 
imagination will be a poor thing if it has to be nurtured on 
the teacher's badly-drawn figures. 



CHAPTER XXII 

Orthographic Projection 

Elementary Work 

Below we reproduce subject-matter suitable for two or 
three preliminary lessons on orthographic projection to the 
Middle Forms. Time can seldom be found for much ruler 
and compass work, but freehand drawings, rapidly executed 



294 



CRAFTSMANSHIP IN MATHEMATICS 



in association with the teacher's own blackboard demon- 
strations, may be made to serve a useful purpose in laying 
the foundations of the subject. Higher up the school, if time 
permits, more advanced work should be taken. It helps the 
ordinary geometry, plane and solid, greatly. 

In preparing drawings for builders, architects make plans 
and elevations of buildings to be erected. A plan of a 
thing is an outline on a horizontal plane; an elevation is an 
outline on a vertical plane. 

Push the table up against the wall. On the table place 
a rectangular block with two faces parallel to the wall. Chalk 

on the table an outline of 
the base of the block, and 
thus make a plan of the 
block. Now push the block 
against the wall, and chalk 
an outline on the wall. 
This is an elevation of the 
block; the elevation a'b'c'd' 
in the figure is a projec- 
tion of the face abed of the 
block. 

It is a very simple kind 
of projection, because most of the work to be done depends 
on the drawing of perpendiculars and parallels. The 
projectors and other working lines are nearly all perpen- 
diculars and parallels. A word implying perpendiculars 
and parallels is " orthographic ", and the projection is some- 
times called orthographic projection. 

As it is not very convenient to draw on the wall, we 
sometimes use two boards hinged at right angles. The next 
figure shows such a pair, first of all hinged in position, then 
unhinged and the vertical plane turned back into the hori- 
zontal. The figure shows two plans and elevations of the hut 
in fig. 136. The first plan shows the long sides of the hut 
parallel to the vertical plane and the elevation a side eleva- 




Fig. 130 



ORTHOGRAPHIC PROJECTION 



295 



tion. The second plan shows the long sides perpendicular 
to the vertical plane, and the elevation an end elevation. 
The term front elevation is also sometimes used. An ele- 
vation is often spoken of as a view. 

An architect would not draw two plans of one building, 
but he would always draw two or more elevations, in order 





Fig. 131 

to make the builder understand exactly what the building 
was to be like. 

Note that all the projectors (shown as broken lines) are 
perpendiculars and parallels. 

The two boards are shown hinged merely to help you to 
understand how the plan and elevation are related, but plans 
and elevations are commonly drawn as in fig. ii. An architect 
would not trouble to draw the outlines of the two boards. 
He just rules a line across the paper (marking it XY some- 
times), draws the plan below, and the elevation above. More 
frequently than not, he uses separate pieces of paper, but 
he always remembers how the separate drawings are related. 
The XY line is sometimes called the ground line: it is the 
line of contact of the vertical plane with the ground. It is 
usual to keep the plan a little distance away from this line, 
but to let all elevations stand on the line. 

Although your plans and elevations will always be drawn 
on the flat, you will sometimes find it useful to fold your 
paper at right angles on the XY line, and to place the object, 
if small enough, in position on the horizontal plane. You 
can then see more plainly what the elevation on the vertical 



296 



CRAFTSMANSHIP IN MATHEMATICS 



plane will be. For instance, in the figure at the beginning 
of this section, suppose a pencil is placed in contact with 
the edge ad of the block, and the block removed. You can 
see at once that the plan of the pencil is bc y and that the 
elevation is a'd'. 

Remember that plans and elevations of any object are 
obtained by drawing perpendicular projectors to the 
Horizontal Plane (H.P.) and Vertical Plane (V.P.). The 
feet of these projectors are then joined in such a way that 
the lines correspond to the edges of the object itself. 

Here are some examples of plans and elevations. Copy 
them full size. Then fold your paper on the ground line 
and turn the V.P. into position. 

1. Plans and elevations of a line 3" long. Hold a piece 
of wire or a short pencil in position, so that, looked at from 



x j- 



567 



Fig. 132 



above, it covers the plan, and looked at from the front, it 
covers the elevation. Then, in each case, try to describe 
the position of the pencil with reference to the two planes, 
checking your descriptions by the correct descriptions below. 





i. 


2 


3. 


4. 


5. 


6. 


7. 


To the H.P. the line is 


II 


JL 


II 


60 


|| 


45 


60 


To the V.P. the line is 


II 


II 


J_ 


II 


45 


45 


30 



2. Plans and elevations of a rectangular sheet of paper, 
3" X 2": 



ORTHOGRAPHIC PROJECTION 



297 





The Plane of 
the Paper is 


The Long 
Edges are 




The Plane of the 
Paper is 


The Long Edges are 


1. 


|| to V.P. 


|| to H.P. 


5. 


|| to V.P. 


30 to H.P. 


2, 


II V P 


JL H.P. 





II H.P. 


60 V.P. 


3. 


II ,, H.P. 


II V.P. 


7. 


45 both planes 


|| both planes 


4. 


II H.P. 


JL V.P. 


8. 


-L ,, both planes 


45 both planes 




3. Plans and elevations of a square prism: 




Positions as follows: 

(1) Standing on base, two sides || to V.P. 

(2) Lying on a side, all sides JL to V.P. 

(3) Lying on a side, bases J- to both planes. 

(4) Standing on a base, one diag. of base J- to V.P. 

(5) Lying on a side, two sides 30 with V.P. 

(6) Same as No. 1, with a section AB to V.P. 

(7) Same as No. 5, with a section CD JL to H.P. 



298 



CRAFTSMANSHIP IN MATHEMATICS 



It is sometimes necessary to know the shapes of sections 
of solids. Two are shown above. The cut surfaces are in- 
dicated by cross hatching. When in doubt about such a 
shape, make a rough model and cut through it. 

The positions of objects with respect to the two planes 
may be described in more than one way. In the first of the 
last series, for instance, we might have said two sides JL to 
V.P. In the third, we might have said two sides || to the 
V.P. and two || to H.P. But the description must always be 
sufficient to fix the object in a particular position. 

4. Plans and elevations of other solids: 





Fig. 135 



Position of solids: 



(1) Hexagonal prism standing on base, two sides || to V.P. 

(2) Hexagonal prism lying on a side, long edges 30 to V.P. 



ORTHOGRAPHIC PROJECTION 299 

(3) Hexagonal pyramid standing on base, two edges of 
base || to V.P. 

(4) Cylinder standing on base, with section AB -L to H.P. 

(5) Cylinder lying down, axis 30 to V.P. 

(6) Cone standing on base, with section CD -L to V.P. 

Sections which are cut obliquely to one of the two pro- 
jection planes may give a little trouble, especially if their 
shapes cannot be first imagined. The shape of an oblique 
section through a cylinder may be shown by half filling a 
round bottle with water and holding the bottle obliquely; 
the water-surface gives the shape of the section an ellipse. 
So with a square bottle, or a conical flask. Or you may push 
the solid obliquely into the ground, down to the level of the 
section line. The shape of the mouth of the hole is the shape 
of the section. 

More Advanced Work 

Here are types of problems suitable for more advanced 
pupils: 

1. Determine the projection of three spheres of different 
radii, resting on the ground in mutual contact. 

2. Determine the projections of the curve of intersection 
of a cone penetrating a cylinder, the axes of the two solids 
intersecting at a given angle. 

3. Determine the shadow cast by the hexagonal head of 
a bolt with a cylindrical shaft, the bolt standing vertically 
on its screw end, from given parallel rays. 

4. Determine the shadow cast by a cone standing on the 
ground, the direction of the light being so arranged as to 
throw part of the shadow on the vertical plane. 

For shadow-casting problems, it is a good plan to place 
the object in strong sunlight, so that the shadow can actually 
be cast on the horizontal plane (and vertical plane, too, if 
necessary), and examined. The problems then become very 



300 CRAFTSMANSHIP IN MATHEMATICS 

simple. Shadows cast by artificial lights are less serviceable, 
since the light rays are necessarily not parallel. 

As a rule there is no time for, and there is very little point 
in, making projections of groups of objects, but cases of simple 
interpenetrations make such good problems that one or two 
are worth doing. 

Speaking generally, the ground covered in orthographic 
projection should be enough to enable Sixth Form boys 
to solve, readily and intelligently, such stock theorems of 
projection as these: 

The projection on a plane of an area in another plane; 
and particular cases, e.g.: 

(a) Projection of an ellipse into a circle, and the ratio 
of their areas. 

(f$) The projective correspondence between the per- 
pendicular diameters of a circle and conjugate diameters in 
an ellipse. 

(y) Extension of the properties of polars from the circle 
to the ellipse. 



CHAPTER XXIII 

Radial Projection 

First Notions 

For Sixth Form boys learning mathematics seriously, a 
knowledge of radial projection is at least as important as a 
knowledge of orthographic projection. Here is the sub- 
stance of a lesson for beginners: it is the sort of lesson one 
might expect to hear an intelligent art teacher give. 

Stand about 18" or 24" from a window, keeping your 
head perfectly steady, and, with a piece of wet chalk, trace 
accurately on the glass an outline of a distant building. 



RADIAL PROJECTION 301 

When you have finished, it is easy to imagine straight threads 
passing from all the principal points in the building, through 
the corresponding points in your sketch on the glass, to 
your eye. Every line in the sketch exactly covers the corre- 
sponding line in the building. The drawing is another kind 
of projection. But the projectors are no longer perpen- 
diculars; they all radiate from your eye, and they all pass 
through the vertical plane on which you have made the 
sketch, to the building. The vertical plane on which you 
have made the picture is called the picture plane. This 
kind of projection is called radial projection or perspective 
projection. Perspective drawings are the kind of drawings 
made by artists. Pictures are painted in accordance with 
the rules of perspective. The camera also follows these 
rules. Pictures and photographs represent things as they 
appear to the eye. 

Here is the perspective projection of a hut: 




In the hut itself, the three vertical lines, AB, CD, EF, are 
all equal. In the drawing, CD, the one nearest the observer, 
is the longest, and those farther away are shorter. So with 
the verticals in the doors and in the windows. All parallels 
which recede from the observer seem to approach each other, 
and at last to meet at a point on a line level with the eye. 
Equal parts of a horizontal in the object are unequal in the 
drawing (compare the horizontal window-bars in the two 
windows). The farther away a thing is taken, the shorter it 
becomes in the drawing. If you have made an accurate 
chalk-drawing on the window, you can teach yourself a good 
deal about perspective. 



302 



CRAFTSMANSHIP IN MATHEMATICS 



If, however, your chalk-drawing is not satisfactory, do 
this instead. Take a rectangular wooden frame of some sort 
(an old picture-frame will do), say about 15" X 10". Drive 
in tacks two-thirds of their length, at equal distances apart, 
say 1", all round the edge. Stretch cotton across the frame 
and round the tacks in such a way as to divide up the frame 
into squares. Now divide up a piece of drawing paper into 
the same number of squares. Place the frame in a vertical 
position between your eye and a suitable object or view 
that may be sketched. If you sketch a house (a very suitable 
object) get back far enough to see the whole house easily 
within the frame. Now observe what part of the object 
appears within a particular square of the frame, and sketch 
that part in the corresponding square on your paper. And 
so on. With care you may make a fairly accurate drawing, 
and can then learn a good deal about perspective, more 
particularly about converging lines and diminishing lengths. 
You may also learn much from a large photograph of a build- 
ing, especially if you can compare the photograph with the 
building itself. 




Here is a perspective sketch of three bricks in a row. It 
is as they would appear in a photograph. The middle brick 




Fig. 138 



is in the middle of the picture, and the photographer points 
his camera towards it. If he were photographing, say, brick 
C alone, he would turn his camera round and point towards 



RADIAL PROJECTION 

that brick. His picture would 
then be like D. Neither A nor C 
is the correct drawing of a brick 
unless the brick is to the left or 
right of a group of things, as in 
fig. 137. 



303 



The Picture Plane. 
Models 



Use of 



The ordinary perspective text- 
book prepared for Art teachers is, 
generally, just a book of rules, 
rules with only the faintest tinge 
of mathematics in them. I have 
known boys make faultless and 
most elaborate perspective draw- 
ings of groups of objects in dif- 
ferent positions, and yet they have 
had the most hazy ideas of the 
inner nature of the rules they have 
been applying. And yet, at bottom, 
the whole thing is a study, and a 
simple study, too, of similar tri- 
angles. 

This figure shows the Picture- 
plane 12 ft. from the observer, 
with his eye 12 ft. away and 5 ft. 
above the ground at S. P. (his 
Station Point). The Picture-plane 
meets the ground in the ground- 
line. The point on the Picture- 
plane immediately opposite the 
eye is the Centre of Vision. The 
horizontal through the Centre of 
Vision is called the Horizontal 
Line. Radial projectors run from 




304 



CRAFTSMANSHIP IN MATHEMATICS 




the eye to each corner of a block fixed behind the Picture- 
plane and cut the Picture-plane in points which, when 



RADIAL PROJECTION 305 

joined up, give on the Picture-plane a perspective picture of 
the block. 

Fig. 140 shows the sort of perspective drawing that appears 
in the textbooks. The pupil must see the relation between 
figs. 139 and 140. In fig. 139 the line from the eye to the 
C. of V. is represented at right angles to the P.P. That line 
must be supposed to be hinged at C. of V. and to turn 
on the hinge through 90 until it comes into the same plane 
as the P.P., as in fig. 140, which represents a drawing in one 
plane, the plane of the paper. It is imperative that the boys 
see fig. 139 as a model. Only then will they be able to under- 
stand fig. 140 completely. Then the points of distance, 
vanishing points, and measuring points are all a matter of 
very easy geometry. 

In practice, it is an advantage to substitute for the glass 
P.P. a sheet of perforated zinc, or a square of stretched 
black filet net of T V' mesh, and to run threads (fastened 
with drawing pins to the corner of the rectangular block or 
other object being sketched) through the appropriate holes 
in the zinc or net to the ring representing the eye, where 
they may be secured. A drawing may then be represented 
in threads of another colour, run from hole to hole in the 
zinc or net, instead of in chalk as when glass is used. 

Main Principles 

The main principles of perspective, mathematically 
considered, are all reducible to a small handful of three- 
dimensional problems. One will suffice to illustrate the 
degree of difficulty. 

Given any point on the ground-plane ', to determine its 
position on the picture-plane. 

Since solids are determined by planes, planes by lines, 
and lines by points, it will suffice to determine the position 
in the picture-plane of just one point on the ground-plane. 
This really solves the general problem, inasmuch as any 
other point may be similarly determined. 

(E291) 31 



306 CRAFTSMANSHIP IN MATHEMATICS 

Let M be the point on the ground-plane. Drop a per- 
pendicular MN on the picture-plane, and another EC from 
the eye E to the C. of V. on the P.P. Since EC is parallel 
to MN, both EC and MN, and also CN and ME, are in the 
shaded oblique plane. NC is the complete projection of 
the line NM extended to an unlimited length behind the 




Fig. 141 

P.P.; and the point R, where ME intersects CN, is the pro- 
jection of M. Hence R is the point to be determined. 

Now in the oblique plane we have the two similar triangles 

NR MN 

RCE, RMN, Hence = tEl 9 i.e. CN is divided at R 
. , . RC CE 

in the ratio: 

distance of point M from P.P. 
distance of observer from P.P. 

Thus CN being drawn, 
R can be determined at 
once. 

Suppose M is 3' 
behind the P.P. and the 
observer is 10' in front. 
It is required to divide 
CN in the ratio 3 : 10. 

Through C and N draw any pair of parallels. Measure off 
NX equal to 3 units and CY equal to 10 units. We have 




N 



Fig. 142 



RADIAL PROJECTION 307 

. ., . , NR NX 3 . . 

two similar triangles. Hence ~ = = , i.e. the 

position of R in CN is determined. 

Hence if the space between the two parallels represents 
the P.P., if C is the C. of V., and if CN is the complete projec- 
tion of a line perpendicular to the picture-plane and meeting 
it in N, the projection of any point in this perpendicular line 
may be found by the above method. 

In practical perspective we use as a pair of parallels the 
horizontal eye-line and the ground line. This is a mere 
matter of convenience; any other pair of parallels drawn on 
the P.P. would do equally well. 

Measuring points, vanishing points, and the rest are all 
determined by the consideration of virtually the same prin- 
ciple. In fact the complete art of perspective projection 
lies in that principle. With the model, the whole thing be- 
comes simplicity itself. The perforated zinc (or net) P.P. 
with strings passing through to the eye, and the projection 
of the figure threaded in with threads of a different colour, 
make the main principles so clear that there is little need 
for formal demonstration. The similar triangles then in 
situ tell the whole story. 

Sixth Form Work 

When these main principles underlying the practice of 
perspective have been mastered, the subject should be 
followed up in the Sixth by a few of the stiffer propositions 
associated with the general theory of perspective, treated 
formally and deductively, more especially those concerned 
with triangles in perspective, so far as these are necessary 
for the understanding of the chief properties of the hexastigm 
in a circle; at least Pascal's theorem should be known, though 
as a mere fact in practical geometry this theorem should be 
known lower down the school; its later theoretical considera- 
tion is always a delight to the k^en mathematical boy. 

But to attack such theorems of perspective before some 



308 CRAFTSMANSHIP IN MATHEMATICS 

understanding of the practice of perspective has been acquired 
is to attack theorems that are lifeless. 



CHAPTER XXIV 

More Advanced Geometry 

A Possible Outline Course 

What is sometimes called " Modern " Geometry or 
" Pure " Geometry usually occupies a subordinate position 
in Sixth Form work. This is to be regretted. 

It may be readily admitted that analysis is a powerful 
instrument of research, and doubtless for this reason alone 
mathematicians have given it a very important place in 
recent years. Accordingly, Sixth Form boys tend to devote 
much time to preparation for the work of that kind which is 
demanded of them at the University. But it cannot be denied 
that an intimate acquaintance with geometry is only to be 
obtained by means of " pure " geometrical reasoning. In 
the classroom no branch of mathematics is so productive 
of sound reasoning as is pure geometry. The ordinary 
geometrical theorem admits of a simple, rigorous, and com- 
pletely satisfactory proof, a proof that is convincing and not 
open to question. An elementary knowledge of the properties 
of lines and circles, of inversion, of conic sections, treated 
geometrically, of reciprocation, and of harmonic section, 
ought to be expected from all Second Year Sixth Form boys. 
Many boys now leave school without any conception of some 
of the remarkable properties of the triangle and circle; and 
this ought not to be. 

There is much to be said for beginning with rectilineal 
figures, including a fairly complete study of the tetragram 
and tetrastigm, the more elementary properties of the polygram 



MODERN ADVANCED GEOMETRY 309 

and polystigm, and then for proceeding with harmonic section. 
The remaining topics follow simply. We outline for teaching 
purposes one or two of the different subjects. 

The Polygram and Polystigm 

These may be regarded either as systems of lines inter- 
secting in points, or as systems of points connected by straight 
lines. The simplest figure is that determined by 3 lines or 
3 points. If we have any 3 lines which are not concurrent, 
or if we have any 3 points which are not collinear and which 
may therefore be connected by 3 straight lines, we have 
two systems which are virtually the same, and we may give 
the name triangle to either. 

But with more than 3 lines or points, the resulting figures 
though closely related are not identical. 

Rectilineal figures regarded as systems of lines are called 
polygrams; as systems of points, polystigms. 

A tetragram in its most general form is a complete recti- 
lineal figure of four lines > 
no 3 of which are con- 
current, and no 2 parallel. 
Each line is therefore in- 
tersected by the other 3. 
If the lines be named a, 
b t c, d, their points of in- 
tersection may be named 
by combining the 2 letters 
which denote the inter- 
secting lines. Since there ' Fig. i 43 
are 3 points of intersec- 
tion in each of the 4 lines, we seem to have 12 points of 
intersection in all, but these are reduced to 6, since each 
is named twice. The 6 points of intersection are called 
vertices. 

A tetrastigm in its most general form consists of four 
primary points, no 3 of which are collinear and which do 




310 CRAFTSMANSHIP IN MATHEMATICS 

not fall in pairs in parallel lines. If the points be named 
A, B, C, D, their connectors may be named in the usual way, 
AB, BC, &c. Since there are 3 connecting lines terminating 

in each of 4 primary points, we seem 
to have 12 connecting lines in all, 
but these reduce to 6, since each is 
named twice. The 6 lines are called 
connectors. 

From suitable figures, the number 
of vertices and connectors in the 
pentagram and pentastigm is seen to 
Fig. i 44 be 10, and in the hexagram and hexa- 

stigm, 15. 

We infer that in a polygram of n lines, and in a polystigm 
of n points, the number of vertices and connectors are, re- 
spectively, -~F-^; for the tetragram and tetrastigm give us 
^-~ > , the pentagram and pentastigm '^p, and the hexagram 
and hexastigm *-~. 

In a tetragram a diagonal may be drawn from each of 
the vertices to another vertex; the 6 diagonals reduce to 3. 






Tetragram, with its 3 diagonals Tetrastigm, with its (4 primary and) 

Fig. 145 3 secondary points 

In a tetrastigm y each of the 6 connectors can intersect 
another connector at a point other than the 4 primary points; 
the 6 reduce to 3. These 3 new points are called the secondary 
points of the tetrastigm. 

From suitable figures the number of diagonals and 



MODERN ADVANCED GEOMETRY 311 

secondary points in the pentagram and pentastigm is seen 
to be 15, and in the hexagram and hexastigm, 45. 
Since 

4x3x2x1,- 5x4x3x2 AK 6x5x4x3 

& = 15 = 45 = 

8 8 8 

we infer that in a polygram of n lines, and in a polystigm 
of n points, the number of diagonals and of secondary points 
respectively is *("-i)(*- 2) <"-*>. 

The pupil should check for the pentagram and pentastigm. 
The figures for the hexagram and hexastigm are complicated 
and their analysis is hardly worth while. The pupil should 
note that a polygram and polystigm of the same order are 
reciprocal figures; they give us an excellent example of the 
principle of duality. 

Derived Polygons 

The pupil may be encouraged to establish, from an ex- 
amination of a few particular cases, the principle that the 
number of derived polygons from a polygram or polystigm 
is fe^L'. 

For instance, the number of derived tetragons from a 
tetragram or tetrastigm is 3x ^ x 1 = 3; of derived pentagons 
from a pentagram or pentastigm is 4x3 * 2x 1 12; of hexa- 
gons, 60; and so on. 

Here is a tetragram and its 3 derived tetragons: 

at> 




A tetragram consists of 4 lines with 6 consequent vertices, 
and 3 vertices lie on each of the 4 lines. But in a tetragon 



312 CRAFTSMANSHIP IN MATHEMATICS 

there are only 2 vertices in a line, viz. those at the extremities 
of the line; there are thus 4 vertices in all. Hence, for a 
derived tetragon, we have to select 4 vertices out of the 6, 
in such a way that 2, and 2 only, may lie on and determine 
the extremities of each of the 4 lines. Such a selection is 
known as a complete set of vertices for a derived tetragon. 
Note that, whatever vertex is chosen as a starting-point, that 
vertex must be the point where the figure is completed. 
Here is a tetrastigm and its 3 derived tetragons. 




Fig. 147 

Analogous reasoning applies. We have to select 4 connectors 
out of 6, in such a way that 2, and only 2, may terminate in 
each of the 4 vertices. The selection is known as a complete 
set of connectors for a derived tetragon. 

The boys may be given the task of drawing the 12 
pentagons from a pentagram and the 12 from the pentastigm. 
But they must set to work systematically or there will be 
confusion. Consider the pentastigm with its 5 primary 
points, A, B, C, D, E. Select AB as the initial connector. 
Associated with it as a second connector we may have BC, BD, 
or BE; we then have the first two connectors formed in 3 
different ways, viz. AB, BC; AB, BD; AB, BE. The first 
two connectors being fixed, the remaining 3 can be selected 
in 2 different ways, and thus we have 6 different pentagons 
formed with AB as a first connector. Now do exactly the 
same thing with the other 3 connectors terminating in A. 
And so on. 

Do not forget that the polystigm is the key to many of 
the mediaeval tree-planting problems. Given n trees, what 
is the greatest number of straight rows in which it is possible 



MODERN ADVANCED GEOMETRY 



313 



to plant them, each row to consist of m trees? For instance, 
given 16 trees, plant them in 15 rows 
of 4. 

Construct a regular pentagram with 
such of its diagonals as are necessary to 
form an inner second pentagram. The 
introduction of these diagonals gives 6 
new points, which, with the 10 vertices 
of the pentagram, make 16 points. 

The general problem has never 
been completely solved. 




Harmonic Division * 

The Pythagoras relation, golden section, and harmonic 
division, are the 3 keys of pure geometry, yet harmonic division 
frequently receives but very scant attention. The principle 
itself once fully grasped, the actual proofs of theorems in- 
volving it are generally of the simplest. 

The approach to the subject and its problems may be 
effectively made in this way: 

(1) Divide a line internally and externally in the same ratio, 




Fig. 149 



say 5:2. Note that the correct reading of the ratios is from 
the extremities of the line to the point of division; thus for 



In speaking of cross-ratios, avoid the term " anharmonic ", since it implies 
" not harmonic ", whereas a cross-ratio may be harmonic, for it may be the cross- 
ratio of an harmonic range. 



314 CRAFTSMANSHIP IN MATHEMATICS 

internal division we have ~ ; for external division -. Also 

BPj BP 2 

note the sign as well as the magnitude; e.g. for internal 
division, AP l and BP X are measured in opposite directions 
and the ratio is therefore negative; for external division 
AP 2 and BP 2 are measured in the same direction, and the 
ratio is therefore positive. Give important examples of this, 
for instance the theorems of Ceva and Menelaus. 

(2) Algebraic Harmonic Progression. Definition: If a, 

, , . TT , a b a 1.12 

0, and c are in H.P., then = -; or, - + - -; or 

b c c a c b 

h 

a -f c' 

(3) Compare the geometry and the algebra. A line AQ 
is said to be " harmonically divided " at P and B when, if 
AQ = a, AB = b 9 AP = c, a, b, and c are in H.P. 

c . a - b a /t , c . . , . BQ AQ 

Since = - (by definition), /._=:_, 

b C C JjJr Ar 

or AQ X BP = AP X BQ; (Cf. fig. 150.) 

i.e. product of whole line and middle segment equals product 
of external segments. Hence if AB is divided harmonically 



A I 1 ! 19 

|^ .j-a-J *, 

j^_ b 1 -^j 

K--c H 

Fig. 150 

at P and Q, PQ is divided harmonically at A and B. In other 
words, AB is divided internally and externally at P and Q 
in the same ratio; and PQ is divided internally and externally 
at B and A in the same ratio. 

(4) Harmonic Ranges. If a line AB is divided harmonically 
at P and Q, the range of points {AB, PQ} is called a harmonic 
" range ". The pair of points A and B are said to be con- 
jugate to each other; so with the points P and Q. We may 



MODERN ADVANCED GEOMETRY 315 

conveniently name a harmonic range thus (AB, PQ}, the 
comma being inserted to distinguish the pairs of conjugate 
points. 

(5) Harmonic Pencils. Define " ray " and " pencil ". 
Every section of a harmonic pencil is a harmonic range, e.g. 
(AB, PQ), (A'B', P'Q'). A pencil O.APBQ is harmonic if 

C> Q 





Fig. 151 



a transversal MN parallel to one ray OQ is bisected by the 
conjugate OP. 

A range may be read {AB, PQ} or (APBQ), and a pencil 
may be read O(AB, PQ) or O.APBQ. Adopt one plan and 
adhere to it, or the boys may be confused. It is a good plan 
to use coloured chalks for every harmonically divided line 
on the blackboard, and always of the same colour. Harmonic 
division is so useful that its immediate recognition is desirable. 

The Complete Quadrilateral 

The commonest theorems involving harmonic section 
concern (1) the complete quadrilateral, (2) pole and polar. 

Fig. 152 (i) shows a tetragram with its 3 diagonals (2 pro- 
duced to meet) which are indicated by heavy lines. Each of 
the 3 diagonals is harmonically divided by the other two. Fig. (ii) 
shows a tetrastigm with its 6 connectors also indicated by 
heavy lines, and with lines (faint) joining the secondary 
points in pairs. Each of the 6 connectors is harmonically divided 
by (1) the secondary point through which it passes, and (2) 
the line joining the other two secondary points. If we superpose 
one figure on the other, we get a remarkable series of harmonic 



316 CRAFTSMANSHIP IN MATHEMATICS 

pencils, of mutual harmonic intersections, and of collinearities 
Actually, of course, they can be picked out in fig. 152 (ii). 




(0 

Fig. 152 

If the complete quadrilateral is approached in this way, 
the boy's interest and curiosity is aroused. He is greatly 
surprised to discover that a simple thing like a quadrilateral 
has so many remarkable properties. 

But the important thing is for the boy to realize that this 
general quadrilateral generalizes the theorems of all particular 
quadrilaterals. For instance, suppose that the line EF in the 
last figure is removed to an " infinite " distance from C, the 
4 points C, A, D, B, become the vertices of a parallelogram; 
and since R is the harmonic conjugate of the point in which 
CD intersects EF with respect to the points C and D, R 
becomes the middle point of CD. Thus the theorem of 
the complete quadrilateral is a generalization of the theorem 
that the diagonals of a parallelogram bisect each other. 

Impress on the boys the importance of the use of a general 
figure in their geometrical work, and the fact that from its 
properties the special properties of a particular figure may 
often be inferred. 

Pole and Polar 

The dark lines in the figures represent harmonically 
divided lines: (i), (ii), every chord which passes through 
the pole P is cut harmonically by the polar; (iii), if the polar 
of PX passes through P 2 , the polar of P 2 passes through P x . 
In all such cases, let the pole and the corresponding point 



MODERN ADVANCED GEOMETRY 



317 



on the polar be consistently marked, say, by P's. Also let the 
circumference be cut in Q's. A consistent system of naming 
all harmonically divided lines is a great advantage. 




(0 



For a good general problem on harmonic section, see 
Scientific Method, pp. 387-9. 



Concurrency and Collinearity 

Many theorems involving these principles form excellent 
practical problems for careful work in junior classes. En- 
courage young boys to bisect the angles of a triangle, to bisect 
the sides, to draw perpendiculars from the mid-points, &c., 
and to make discoveries for themselves. Let them thus obtain 
the facts. A little later, the simpler theorems and their proofs 
may be given; e.g. concurrent lines through the vertices 
of a triangle, the medians, the perpendiculars to the opposite 
sides, two exterior bisectors and the internal bisector of the 
three angles. A little later still: given the Menelaus relation, 
prove that the points are collinear; given the Ceva relation, 
prove that the points are concurrent. Pascal's and Brianchon's 
theorems will, of course, always be included. Another good 
type of theorem is this: four points on a circle and the tangents 
at those points form, respectively, two quadrilaterals whose 
internal diagonals are concurrent and form a harmonic pencil, 
and whose external diagonal points are collinear and form 
a harmonic range. The principles of concurrency and col* 
linearity are so important that they cannot be too strongly 



318 . CRAFTSMANSHIP IN MATHEMATICS 

emphasized, but with most boys facility comes only after 
much practice with varied types of problems. 

Pascal's theorem suggests the study of the hexastigm, of 
which that theorem is the simplest property. The theorem is 
usually quoted, " The opposite sides of any hexagon in- 
scribed in a circle intersect in 3 collinear points ", but a 
more precise statement is, " The 3 pairs of opposite connectors 
of a hexastigm inscribed in a circle intersect in 3 collinear 
points ". Fully expressed, this comes to, " The 15 connectors 
of a hexastigm inscribed in a circle intersect in 45 points 
which lie 3 by 3 on 60 lines ". I have seen one passable 
figure prepared by a boy; he was looked upon as the fool of 
his Form, though he was extraordinarily successful in the 
use of ruler and compasses. Elaborate drawing of this kind 
is largely a waste of time, and, after all, the hexastigm still 
remains to be investigated fully. 

The Further Study of the Triangle and Circle 

There are numerous theorems on the triangle, many of 
them simple, many useful, many beautiful. For instance, 
those concerning triangles in perspective, pole and polar with 
respect to a triangle, symmedian points of a triangle, Brocard 
points of a triangle. 

So with the circle: the nine-point circle, escribed circles, 
the cosine circle, the Lemoine circle, the Brocard circle. 

It is important to leave on the boy's mind a vivid im- 
pression of the remarkable properties that even now are 
frequently being discovered concerning the triangle and 
circle. Boys who, when they leave school, know no more 
pure geometry than that contained within the limits of 
School Certificate requirements are certainly not likely to 
devote leisure moments to a systematic playing about with 
circles and triangles, in the hope of hitting on some new and 
perhaps remarkable property yet undiscovered. 



MODERN ADVANCED GEOMETRY 319 

Conic Sections 

The pure geometry and the algebraic geometry of the 
cone should be studied side by side. If either has to be 
sacrificed, let it be the latter. Algebraic manipulation is all 
very well, but the cone is a thing which occupies space, and 
when its spatial relations are reduced to symbols, these 
symbols may assume, in the pupils' minds, an importance 
which is not justified, and the geometry proper may be 
overshadowed. 



CHAPTER XXV 

Geometrical Riders and their Analysis 

In those schools where riders are, as a rule, solved readily, 
schools where boys take a real delight in attacking new ones, 
the secret of success seems to be that right from the first 
every new theorem and every new problem has been presented, 
not as a thing to be straight-away learnt, but as a thing to 
be investigated and its secret discovered. The boys do not 
learn a new theorem or problem until they have been taught 
how to analyse it, and to discover how it hangs on to what 
has gone before. 

General instructions should include advice as to the 
necessity of drawing a general figure, of drawing that figure 
accurately, and of setting out definitely what is " given " 
and what is to be proved. We append a few instances of 
problems and theorems actually solved in the classroom, 
with a brief summary of the sort of arguments used. 

1. O is the mid-point of a straight line PQ, and X is a point 
such that XP = XQ. Prove that the Z.XOP is a right angle. 



320 CRAFTSMANSHIP IN MATHEMATICS 

We argue in this way: 

(1) What facts are given? 

OP = OQ, 
XP = XQ. 

(2) What have I to prove? 

That ZXOP is a right angle. 

(3) Since I have to prove that Z.XOP is a rt. 

join XO. 

x 



I must 




(4) How have I been able to prove before, that an L is 

a it. Z? 

(i) Sometimes by finding it to be one of the two eq. 
adj. Z.s making a str. L. 

(ii) Sometimes by finding it to be an angle in a semi- 
circle. 

(iii) Sometimes by finding it to be at the intersection 
of the diag. of a sq. or a rhombus, 

(5) The first of these looks possible here. Are the adj. 

/.s at O equal? Yes, if the two As XOP and 
XOQ are congruent. 

(6) Are these As congruent? Yes, three sides in the one 

are equal to three sides in the other, as marked. 

Now I know how to write out the proof, in the ordinary 
way: I therefore begin again, and make up a new figure as 
I proceed. 



GEOMETRICAL RIDERS 321 

Proof. Join XO. In the As XOP, XOQ, 

OP == OQ, (given) 
XP - XQ, (given) 
XO is common, 

.-. A XOP = A XOQ; (three sides) 
i.e. the two As are equal in all respects. 
.-. ^XOP- ZXOQ, 
.- . ^XOP == 90. (half the str. POQ) 

(which was to be proved) 

2. ABCD is a parallelogram; E is the mid-point of BC, 
and AE and DC produced intersect at F. Prove that AE = EF. 




Argument. 

(1) What facts are given? 

(i) ABCD is a /Z7m; . . its opp. sides are ||. 
(ii) BE == EC (by constr.). 

(2) What have I to prove? 

That AE - EF. 

(3) How have I been able to prove before, that two lines 

are equal? 

(i) Sometimes by finding them in two congruent As. 
(ii) Sometimes by finding them in a A with two angles 

equal, 
(iii) Sometimes by finding them to be the opp. sides 

of a H7m. 

(B291) 22 



322 



CRAFTSMANSHIP IN MATHEMATICS 



(4) Does either of these plans seem possible, to prove 

AE eq. to EF? 

(5) Yes, the first, for the As ABE and FCE look congruent. 

(6) Are they congruent? 

(7) Yes. Two Zs and a side, as marked. 

Now I know how to write out the proof in the ordinary way 
Proof. In the As ABE, FCE, 



BE = EC, 
ZAEB = ZFEC, 
ZABE = ZFCE, 
A ABE -A FCE, 
.-. AE-EF. 



(constr.) 

(vert. opp. Ls) 

(alt. Ls; BC across \\s AE, DF) 

(2 Z.s and a side) 

(which was to be proved) 



3. Draw a circle of |" radius to touch the given line AE 
and the given circle CDE. (The given line must not be more 
than 1" from the given circle.) 




(*> 



Fig. 156 



This is a problem, and we have to discover the method 
of construction. I assume the problem done, and I make a 
sketch of the required circle in position, as accurately as 
possible (fig. 156, i). I examine the figure, and I observe that 
the line QR to the pt. of contact R = tf and is AB; that 
HQ = HP + Y and passes through the pt. of contact P. 

Argument. 

(1) Since the required O has to touch the line AE> its 



GEOMETRICAL RIDERS 323 

centre must lie somewhere on a line FG || AB, and 
Y from AB (fig. 156, ii). 

(2) Since the required O has to touch the O CDE, its 

centre must lie somewhere on the circle LMN 
having the same centre as CDE and having a radius 
HQ equal to radius HP + (fig. 156, ii). 

(3) Since the centre of the required O lies both on the 

line FG and on the circle LMN, it must be at a 
point of intersection of FG and LMN. 

Now I know how to construct the circle. 
Construction. 

(1) Draw a line FG || AB, -|" away from it. 

(2) From centre H, with radius equal to HP + i"> draw 
O LMN. 

(3) From one of the pts. of intersection of this line and 
circle, say Q, as centre, draw a circle RSP of |" radius. This 
is the required O. 

Proof. 

(1) The circle RSP has a radius of |". (constr.) 

(2) The circle touches AB (in R), for any circle of \* 
radius having its centre on the line FG must touch AB, a 
from the centre Q passing through the pt. of contact. 
(constr.) 

(3) The circle touches the given circle CDE (in P), for 
any circle of J" radius having its centre on the circle LMN 
must touch CDE, the line joining the centres passing through 
the point of contact, (constr.) 

(4) Therefore the circle is constructed in accordance 
with the given conditions, (which was to be done). 

4. From the right angle of a right-angled triangle, one 
straight line is drawn to bisect the hypotenuse, and a second 
is drawn perpendicular to it. Prove that they contain an angle 



324 CRAFTSMANSHIP IN MATHEMATICS 

equal to the difference between the two acute angles of the 
triangle. 

A 





BED c 

Fig. 157 

Given. 

(1) a rt. L at A. 

(2) rt. Zs at E. 

(3) DB = DC. 

(4) DA - DB DC. (Since a O will go round the 

A ABC on BC as diameter.) 

Further, an examination of the figure shows that in the 
2 rt. Zd As BAC, BEA, ZABE is common; .'. the two As 
are equiangular. This fact may prove useful. 

Required to prove: ZEAD ( ZABC - ZACB). 

Argument. This is a type of problem in which we may 
first usefully test a particular case by assigning to some 
angle a number of degrees, and then calculating the 
number in some or all of the other angles. For instance, 
let ZABC- 65 (not 30, or any other factor of 360, lest 
a fallacy creep into our argument). 

If ZABC = 65, ZACB = 25 (the complement). 
If ZACD - 25, ZCAD - 25 (for AD - DC). 
Also ZBAE = 25 (equiangular As, as above). 

Again, if ZABC = 65, ZBAD = 65 (for DB = DA), 
and ZEAD == ZBAD - ZBAE - 65 - 25 = 40. 

But ZABC - ZACB - 65 - 25 - 40, 
/. ZABC - ZACB = ZEAD. 



GEOMETRICAL RIDERS 325 

Thus the theorem is true in this particular case. We are 
therefore now in a position to generalize the result and to 
set out the proof in the ordinary way. 

Proof. 

(1) DB = DA, (given) 

.-. ZDBA- ZDAB. 

(2) BAG and BEA are equiangular rt. Zd As. (given) 
.-. ZACB- ZEAB. 

(3) .-. ZDBA - L ACB = ZDAB - ZEAB (from 1 and 2), 

- ZDAE. 

(which was to be proved) 

5. The figure shows an equilateral triangle ABC within 
a rhombus ADEF, a side of the former being equal to a side 
of the latter. Determine the magnitude of the angles of the 
rhombus. 




Argument. 

From an examination of the figure we know the following 
facts: 

(1) Rhombus: 4 equal sides; opp. sides ||; opp. 2Ls equal. 

(2) Equil. A : sides equal; Z.s equal. 

(3) As ADB, AFC isosceles. 

(4) A EEC isosceles (by symmetry). 



326 CRAFTSMANSHIP IN MATHEMATICS 

All that we know about L magnitudes from the figure are: 

(1) Angle-sum of any A = 180. 

(2) Each L of A ABC = 60. 

We have therefore to try to express the Zs of the rhombus in 
terms of these values. 

From the rhombus, ZFAD + ZADB = 2 rt. Zs. 

Also, ZABD + ZABE = 2 rt. Zs. 

But ZADB = ZABD, (isos. A) 

.-. ZFAD- ZABE. 
Obviously, therefore, ZFAD- ZFED- ZABE- /ACE 

Now the sum of the last 3 of these Zs 

= (sum of Zs of A EEC) + ZABC + ZACB 
= 180 + 120 
- 300. 

= 100; 



.-. ZADE = (180 - 100) = 80. 

The estimate may be set out formally in almost the same 
sequence. 

6. Show that the 4= straight lines bisecting the angles of 
any quadrilateral form a cyclic quadrilateral. 




Fig. 159 



GEOMETRICAL RIDERS 



327 



Let ABCD be the given quadrilateral, and let the bisectors 
of the Zs form the quadrilateral EFGH. 
Let the Zs be marked as shown. 

Given: a = a'; j8 = j8'; y = y'; 8 == 8'. 
Argument, 

If a circle will go round EFGH, the sum of any 2 opp. 

Zs of EFGH 2 rt. Zs; thus a + T = 2 rt. Zs. 
// a + r = 2 rt. Zs, a + j8' + y + 8' = 2 rt. Zs since 

the sum of all the As of the 2 As ABE and CDG 

= 4 rt. Zs. 
But a + /J' + y + 8' we &wo; are equal to 2 rt. Zs, for 

2a + 2)3' + 2y + 28' = 4 rt. Zs (the 4 Zs of the 

quadl.) 
Thus we have found the key. 

Proof. 

The sum of the 4 Zs of the quadl. ABCD = 4 rt. Zs. 

. . the sum of the halves, a + ft' + y + 8' = 2 rt. Zs. 
But the sum of all the Zs of As ABE and CDE = 4 rt. Zs. 
. . a' + T' = 2 rt. Zs, 
.-.a + T = 2 rt. Zs, 
.-. the points E, F, G, H are concyclic. 

(which was to be proved) 

7. Three points D, E, 
and F in the sides of a tri- 
angle ABC are joined to 
form a second triangle, so 
that any two sides of the 
latter make equal angles with 
that side of the former at 
which they meet. Show that 
AD, BE, and CF are at 
right angles to BC, CA, Fig . I6o 

and AB, respectively. (You 
may not assume properties of the pedal triangle.) 




328 CRAFTSMANSHIP IN MATHEMATICS 

Given: a = a'; = '; y = y'. 
Required to prove: AD is -L to BC, &c. 
Argument. Assume that AD is -L to BC. 
Then .* a = a', S = S'. 
DF FG 



= 
FG GE' 

DA 



But this is known, since each ratio = 



AG' 



LThis is easily shown: Produce DE to H; EA is the 
DF DA 

bisector of the external ZGEH of the A GDE; .-.=-= = . 

EG AG 

Similarly, by producing DF to K, it is seen that - . = -_ . 

AG J 



Thus we can make these known ratios our starting-points, 
and set out the proof in the usual way. 

Proof. 

Produce DE to H and DF to K. Then EA and FA are 
the bisectors, respectively, of ext. ZGEH of A GDE, and 
of ext. -dGFK of AGDF. 

DE DF , , DA 
" EG = FG' f r CaCh = AG' 

DE = EG 
" DF FG' 
/. DG bisects ZEDF, 
.'. 8 = 8', 
/. 8 + a = 8' + a', 
.-. AD is to BC. 



GEOMETRICAL RIDERS 



329 



Similarly, we may show that BE -L AC, and CF JL AB. 

(which was to be proved) 

8. Show that the perpendicular drawn from the vertex 
of a regular tetrahedron to the opposite face is 3 times that 
drawn from its own foot to any of the other faces. 

Let ABCD be the tetrahedron, and let AE be the -L from 
the vertex A to the opp. face BCD. Then E is the centroid 
of the ABCD. 

Let a -L EF be drawn to the face ACD; F will meet the 
median AG. 





SECTION THROUGH TETRAHEDRON 
Fig. 162 



To prove: AE = 3EF. 

Argument: Consider the vertical section through ABG. 

We know that, since E is the centroid of BCD, EG = BG, 
Is there an analogous relation between EF and AE? 

If we draw BK 1 AG in the face ACD, BK must be 
equal to AE. 

E F _ EG = i 

BK " BG 3 ' 

/. EF = JBK = |AE 



For the other faces similar results follow from symmetry. 



330 CRAFTSMANSHIP IN MATHEMATICS 

Proof. 

Let a 1 BK from B meet the median AG in K; BK = AE. 
EG = BG. (E is the centroid of BCD) 
EF _EG , 
BK BG *' 

/. EF - PK 

= JAE. (which was to be proved) 

If a rider is in any way of an unusual character, pupils 
sometimes have difficulty in writing out a proof concisely. 
We give an example of an acceptable proof for such a rider. 




Fig. 163 

In a given triangle ABC, BD is taken equal to one-fourth 
of BC, and CE equal to one-fourth of CA. Show that the 
straight line drawn from C through the intersection F of BE 
and AD will divide the base into two parts at G which are 
in the ratio 9 to 1. 

ABEA-3 A EEC, 

AFEA = 3 AFEC, 

.'. ABFA-=3 ABFC, 

= 12 ABFD, 
.'. AF = 12 FD, 
/. AAFC= 12 ADFC 
= 36 ABFD 
= 9 ABFC. 



GEOMETRICAL RIDERS 331 

Now the As AFC, BFC are on the same base FC. Hence the 
vertical height of A AFC above this base = 9 times the 
vertical height of A BFC above this base. 

.'. A AGC 9 ABGC, (on the extended base, GC) 
AG = 9 GB. 

Any reasonable examiner would accept a proof given in 
this form and would be glad to be saved from the trouble of 
reading defensive explanatory matter. 

Books on geometry to consult: 

1. Plane Geometry, 2 vols., Carson and Smith. 

2. Geometry, Godfrey and Siddons. 

3. Geometry, Barnard and Child. 

4. Elementary Concepts of Algebra and Geometry, Young. 

5. Elementary Geometry, Fletcher. 

6. Cours de Ge'ome'trie, d'Ocagne (Gauthier Villars). 

7. A Course of Pure Geometry, Askwith. 

8. Modern Pure Geometry, Lachlan. 

9. Sequel to Elementary Geometry, Russell. 

10. Geometry of Projection, Harrison and Baxandall. 

11. Protective Geometry, Matthews. 

12. An Elementary Treatise on Cross-Ratio Geometry, Milne. 

13. Foundations of Geometry, Hilbert. 

14. The Elements of Non-Euclidean Geometry, Sommerville. 

15. Space and Geometry, Mach. 

16. Analytical Conies, Sommerville. 

17. Curve Tracing, Frost (new edition). An old and faithful friend. 

18. Euclid, 3 vols., Heath. The work on the subject. 



332 CRAFTSMANSHIP IN MATHEMATICS 

CHAPTER XXVI 
Plane Trigonometry 

Preliminary Work 

The pupils' first /acquaintance with the .tangent, sine, 
and cosine should be made during their elementary lessons 
in geometry. Boys soon (learn that the symbols for the 
trigonometrical ratios^ may enter into formulae which can 
be manipulated algebraically; and since, in the algebra 
course, the study of x n and a x is included, it is difficult to 
exclude from it the study of sin x and tan x. Each represents 
a typical kind of function. To each corresponds a specific 
form of curve its own particular picture, the graphic picture 
of the function. Algebra and trigonometry should be much 
more closely linked together, and much of the purely formal 
side of trigonometry might with advantage be sacrificed, 
and greater stress be laid on the practical and functional 
aspects of the subject. The needs of co-ordinate geometry 
and the calculus, of mechanics and physics, should always 
be borne in mind; in fact, much of the work done in trigo- 
nometry might be directed towards these subjects. 

The notion of an angle as a rotating line should be given 
at the very outset of geometry, so that, when in trigonometry 
angles greater than 180 are discovered, the notion will 
already be familiar. The angle of " one complete rotation ", 
and its subdivisions, straight angle, right angle, and degree, 
will, of course, be known, and pupils should be able to draw 
freehand, at once, to a fairly close approximation, an angle of 
any given size, the 30, 45, and 60 angles being quite 
familiar from the half equilateral triangle and the half square. 

Co-ordinate axes and the four quadrants will also be 
familiar from previous work on graphs; so will directed 
algebraic numbers. Angles of elevation and depression will 
already have been measured in connexion with practical 



PLANE TRIGONOMETRY 



333 



problems in geometry and mensuration. Pythagoras should 
be at the pupils' finger-ends; so should the fundamental 
idea of projection} Similar triangles should also be known, 
and ratios of pairs of sides should be equated with readiness. 
Unless all these things are known, really known, the earlier 
work in trigonometry is much hampered by time-consuming 
preliminaries. 

Do not scare the class in the first lesson by hurling at 
their heads all six trigonometrical ratios. Only the tangent, 
sine, and cosine need be studied at first, and these one at a 
time, each as a natural derivative of practical problems of 
some kind. 

The Tangent 

The tangent should come first. Revise a few simple 
geometry problems in heights and dis- 
tances, and let the new trigonometrical 
term gradually replace the geometrical ratio 
which the boys already know. 

We might begin in this way. 

Measure the height of the school flag-staff 
AB. 

Set up the 4' high theodolite at D, at 
a distance of, say, 25' from B, and measure 
the angle AEC (== 58). Make a scale 
drawing. By scale, AC = 40'. Hence AB 
= AC + CB = 40' + 4' = 44'. 




Thus the ratio - ~ = 1-6. 

In other words, when the angle E is 58, AC = 1-6 EC. 
Now look at a series of right-angled triangles with the base 
angle 58. In every case the ratio AC/CE is the same, since 
the triangles are similar. Thus in each case AC = 1-6 EC. 
Hence, whatever the length of EC, we can find the length 
of AC by multiplying EC by 1-6. (Fig. 165.) 



334 



CRAFTSMANSHIP IN MATHEMATICS 



Thus the number 1-6 is evidently associated with the 
particular angle 58. How? It measures the ratio AC/CE, 

i.e. the P^Pendicular of the right-angled triangle AEC. If, 

base 

then, we make a note of this value 1-6, as belonging to the 
particular angle 58, we are likely to find it very valuable 
when dealing with right-angled triangles having an angle 







E c E c E c 

Fig. 165 

of 58; if we know the base we have merely to multiply it 
by 1-6 to obtain the perpendicular.* 

Obviously every angle, not merely 58, must have a special 
value of this kind. We may take a series of right-angled tri- 
angles, with different base angles, say 10, 20, 30, 40, 50, 
60, 70, 80, measure their perpendiculars and bases to scale, 
calculate their ratios, and make up a little table for future use. 

If we liked, we could draw these triangles independently, 
though that would make the arithmetic rather tedious. An 
easier way is to draw a base of exactly 1" in every case; then 
our arithmetic is easy (fig. 166). (Any number instead of 1 
would do, but that would mean a little more arithmetic.) 



58; 



EC -36 






Do not mention the term hypotenuse at all. 
dealt with. 



Let that wait until the sine 



PLANE TRIGONOMETRY 



835 



Mathematicians sometimes make the perpendicular a 
tangent to the circle, fig. 167 (they always remember that 
an angle is concerned with rotation): and for convenience 

they call the ratio P er P^ ndlcu _ lar the tangent of the angle. 

base 

Thus they say, tangent 10 == -18; tangent 20 = -36; and 
so on. They generally write tan for tangent. 




Bi 



-rj- 
84 ; 




Fig. I 66 



Fig. 167 



But remember that the tangent of an angle is just a number 
which shows how many times the perpendicular is as 

perp. 



long as the base; in other words, it is the ratio 



Since 



perp. 

^*- - tan, 
base 



perp. = base X tan\ 



base 
hence in the 



triangle ABC, AC = BC X tan 35, 
i.e. the tan of an angle is the mul- 
tiplier for converting the base into 
the perpendicular. (Fig. 168.) 

There are better ways of finding 
these values than by merely drawing 
to scale; in fact, values to 7 places of 
decimals have been found, the work 
to be done with them (by surveyors, 
for instance) having often to be very accurate. Here is a little 



Fig. 1 68 



336 



CRAFTSMANSHIP IN MATHEMATICS 



table giving the values of the tangents of 10 angles, to 4 places 
of decimals. 



tan 10= 
20 = 
30 = 
40 = 



-1763 
-3640 
-5774 
-8391 



45 = 1-0000 



tan50= 1-1918 
60 = 1-7321 

70 = 2-7475 
80 - 5-671 
89 = 57-29 



373 



y 






































/ 




















/ 


c XT 
















y 





















/ 










-"for 


75- 


373 






__/ 




















/ 


















/ 






| 










^^* 


^ 


/ 








, - 


^- 


^ ' 


^ ' 















l( 


5* 2 


3 


0' 4 

A 


0* 5 


<* 


7 


r a 
75- 


0* 9C 



Fig. 169 

There is no tangent for 90. Can you see why? Can you 
see why the tan of 89 is so large? look at fig. 167. Can 
you see why the tan of 89 59' 59" must be enormously 
large? 

You will remember how, when we had graphed a 
function of #, we were able to obtain other values by 
interpolation. We may do the same with the tan graph; 
in fig. 169, plotted from the above table, you may see 
that the tan of 75 is about 3-73. To get anything 
like accurate values, we should have to have a very large 
graph. 

We give one or two easy practical exercises. 



PLANE TRIGONOMETRY 

A ladder leaning against a house makes an 
angle of 20 with the wall. Its foot is 10' away. 
How high up the house does it reach? 

We have to obtain the height AC, and we 
therefore require to know the tan of the angle B. 
Since A - 20, B = 70. 



= tanB = tan 70 
BC 

= 2-7475 (see table or graph). 
AC ^ BC X 2-74:75 
--- 10' X 2-7475 
= 27-475'. 



Fig. 170 



Two boys are on opposite sides of a flag-staff 50' high. 
Their angles of elevation of the top of the staff are 20 and 
30, respectively. How far are they apart? 




Fig. 171 

Given, length of AB; Required, length of BC and BD. 
Since the angles at C and D are given, we may mark in the 
angles at A. 

Distance of boys apart CD 

= CB + BD 

= AB tan 70 + AB tan 60 
= 50(2-7475 + 1-7321) 
- 223-98 (feet). 

Give ample practice in easy examples of this kind until the 
boys are thoroughly familiar with the fact that the tan is 
just a multiplier, sometimes less than 1, sometimes greater, 

(B291) 23 



338 



CRAFTSMANSHIP IN MATHEMATICS 



for calculating the length of the base from the per- 
pendicular. Vary the exercises, so that the base is not 
always a horizontal. 



The Sine 

To beginners, navigation problems for introducing the 
sine seem to be a little difficult, and may best be taken a little 
later. Here is a suitable first problem. A straight level road 
AB, 20 miles long, makes an angle of 37 with the west-east 
direction AC. How much farther north is B than A? 

In the figure we have to find the length BC. It is easy to 
find this length from a scale drawing: BC =12 miles, i.e. 
B is 12 miles north of C. 





Fig. 172 



G 

Fig. 173 



BC 
BA* 



Now examine the ratio ^. As long as the angle A in 



a right-angled triangle remains 37, the ratio must always 
be the same, no matter what the length of the sides, e.g. 

BC DE FG T - k f r ,. 

p . If then we know the value of this ratio 



for one triangle, we know it for all similar triangles; its 
value is ^j r * 6 - Thus, if AD = 14, DE = -6 of 14 = -84; 
and so on. 

This new ratio is Perpendicular and is called sine. It 

hypotenuse 

is a mere number, and represents how many times the per- 
pendicular is as long as the hypotenuse. We ought really 



PLANE TRIGONOMETRY 



339 



to say, represents what fraction the perpendicular is of the 
hypotenuse, since the value is always less than 1. Thus 
sine 37 = *6 (we generally write sine, sin, though we pro- 
nounce " sin " as " sine "). 

Just as with the tangents, so with the sines: we might 
draw a series of right-angled triangles with base angles 
successively equal to, say, 10, 20, 30, &c., and so construct 
a table. When we constructed fig. 166 for the tangents, we 
made a triangle with a base of 1 unit, because we wanted to 




Fig. 174 

make the arithmetic easy, and then the base was the de- 
nominator of the ratio. In the case of the sine, we will also 
make the denominator of the ratio unity, i.e. we must now 
make the hypotenuse unity. Here is a plan for doing this. 
With O as centre, and unit radius, draw a circle. With the 
protractor, mark in the angles 10, 20, 30, &c.; each radius 
OB, OC, &c., is equal to unity. From the ends B, C, D, &c., 
of these radii, drop perpendiculars to the base, BG, CN, DK, 
&c., and measure them. Since OA = OB = OC (&c.) = T, 
the perpendiculars will be fractions of 1". Now we may 

* . BG -17 . n CN 

obtain the sines: smlO = ~ - == = -17; sm20 = r 

34 
= = -34; sin 30 = '50, &c. By careful measurement, we 



340 



CRAFTSMANSHIP IN MATHEMATICS 



may obtain sines to 2 decimal places. Here is a little table 
to 4 places. 



sin 10 = 
20 = - 
30 = 
40 = - 
45 = 

(, 


1736 
3420 
5000 
6428 
7071 












si 
i 


n50 

60 C 
,70 f 
, 80 C 
, 90 C 


D 
5 
) 
) 
) ' 


766 
866 
939 
984 
L-OOOi 


o 
















, 


^ 


-'"'" 


9 




Sr> 


55- 


1 


82 




/ 


S 






.0 


































/ 


^ 








f 










, 


/ 










^iKiee. ^? 








/ 


/ 


















/ 
















.X 




, 


/ 




















/ 





















y 




















1 


/ 





















10" 20* 30* 4CT 50* 60* 70* 80* 9O* 
65 

ANGLES 

Fig. 175 

By drawing the sine graph, we may obtain the sine of any 
A other angle up to 90, by inter- 
polation; e.g. sin 55 is about -82. 
Remember that the W of an 
angle is just a number. Since 
perpendicular _ ^ . _ 

hypotenuse 

dicular = hypotenuse X sine. Hence 
Fig. .76 " inthetriangleABC,AC=ABsin35, 




PLANE TRIGONOMETRY 



341 



i.e. the sine of an angle is the multiplier for converting the 
hypotenuse into the perpendicular. In this case the multiplier 
happens to be always a fraction. 

Here are one or two easy typical problems: 

A ladder 30' long stands against a vertical wall. It makes 
an angle of 70 with the ground. What is the height above 
the ground of the top of the ladder? (Fig. 177.) 

Given, AB = 30'; ZABC = 70. Required AC. 

AC 

= sin 70 = -94 (from table or graph), 
A. 5 

/. AC = AB x -94 = 30' x -94 = 28-2'. 



A railway slopes at an angle of 10 
for a distance of 1000 yards. What is the 
difference in level of its two ends? (Fig. 
178.) 





Fig. 178 



Given, AB = 1000 yards; Z.ABC = 10. Required AC. 
AC 



AB 



= sinlO = -1736. 



/. AC = AB X -1736 = 1000 yd. X -1736 = 173-6 yd. 



The Cosine 

Projection problems form a suitable beginning. AB 
represents a sloping road 500 yd. long. A surveyor finds that 
it makes an angle of 30 with the horizontal What is the 



342 



CRAFTSMANSHIP IN MATHEMATICS 



projected length on a horizontal line, such as would be shown 
on an ordnance map? 

The projection of a line AB on another line MN is the 
distance between two perpendiculars drawn to MN from the 




M 



Fig. 179 



N 



ends of AB. If MN passes through A, one end of the road, 
only one perpendicular (BC) is necessary. The projection is 
then AC. 

Given, AB = 500 yd.; ZBAC = 30. To find AC. 

From a scale drawing we find that AC 433 yd. 

AC 

Now examine the ratio As long as the angle A in a 

AJJ 

right-angled triangle remains 30, the ratio must always be 

B 




A G E C 

Fig. 1 80 

the same, no matter what the length of the sides, e.g. = -- 
A /-* AJb> AD 

= pgr, for the triangles ABC, ADE, AFG are all similar. If 

then we know the value of this ratio for one triangle, we know 

433 
it for all similar triangles. Its value is or -866. This 

base 50 

new ratio, ^ tenuse > is called the cosine (generally written 



PLANE TRIGONOMETRY 
base 



343 



cos). It is a mere number. Since f = cos, base = hypo- 

hyp. 

tem/5* X cos. Hence, in the triangle ABC, AC = AB cos 30, 
i.e. the cosine of an angle is the multiplier for converting 
the hypotenuse into the base. In this case, again, the 
multiplier always happens to be a fraction. 

Nf 




Just as with the tangent and sine, so with the cosine: 
we may draw a series of right-angled triangles with base 
angles successively equal to say 10, 20, 30, &c., measure 
them up, and so construct a table. And as in the case of the 
sine, we will so construct our triangles that the length of the 
hypotenuse is always unity. 



344 CRAFTSMANSHIP IN MATHEMATICS 

Here is a little table of cosines, to 4 places of decimals: 



cos 10 = -9848 
20 = -9397 
30 = -8660 
40 = -7660 
45 = -7071 



cos 50 = -6428 
60 = -5000 
n 70 = -3420 
M 80 = -1736 
90 = 



By drawing a cosine graph from the above values, we can, 
by interpolation, obtain the value of any other angle up to 
90, e.g. cos 35 = -82 (approx.). 



COSINES '- 



Q 


" **^ 


"\ 


^ 














".A" 


SPS, 


_3S = 


_-Q2 


\ 












O 
.7 








' 


\ 










./ 










\ 


\ 




















\ 








4 














\ 






.X 














\ 


, 




2 
















\ 






















\ 




















\ 




l( 


J Z 


c5 


4 





G 


7 





kO 9( 



55" 

DEGREES 

Fig. 182 

Compare the sine and cosine graphs. Each is an exact 
looking-glass reflection of the other. Now look at che two 
tables of sines and cosines. Each is the other turned upside 
down. Evidently there is a curious connexion between sines 
and cosines. 

It is easy to draw both sine and cosine curves by means 



PLANE TRIGONOMETRY 



345 



of intersecting points made by (1) parallels from an angle- 
divided quadrant, and (2) perpendiculars from the corre- 
spondingly divided abscissa. Note how the two curves 
together form a symmetrical figure, and how they cut in 
one point. What do you infer about this point common 




\ 



O 10 20 30" 40*. 50" 60* 70* 60* 90 

4* 
Fig. 183 

to the two curves? There is evidently some angle the sine 
and cosine of which have the same value. Look at the two 
tables. 

Easy cosine problems. (1) The legs of a pair of compasses 
are 5" long. Find the distance between the points when the 
legs are opened to an angle of 80. 




C 

Fig. 184 



Given: AB = AD = 5"; /.BAD = 80. If AC is the bi- 
sector of /.BAD, ^BAC = 40; hence /.ABC = 50. 



346 CRAFTSMANSHIP IN MATHEMATICS 

Required: length of BD (= 2BC). 

Dp 

r = cos 50; /. BC = AB cos 50 

= 5" X -64, 
.'. BD = 10" x -64 == 6-4". 

(2) C is any point in the line XY. CA and CB are drawn 
on the same side of XY so that CA = 4", CB = 5", LXCA 
= 40, L YCB = 60. Find the projection of ACE on XY. 




^ M C N T 

Fig. 185 

Drop perpendiculars AM, BN, on XY. Then the projec- 
tion of ACB on XY is MN. Required: the length of MN. 

MN = CM + CN 

= AC cos 40 + BC cos 60 
= (4" X -77) + (5" X -50) 
= 5-58". 

Now give the boys the same two problems again, making 
them use the sine instead of the cosine. Hence give them 
the first notion that the sine and cosine are so closely related 
that one may sometimes be used instead of the other. Make 
them remember this: 

If the hypotenuse is given, 

(1) use the sine to find the perpendicular; 

(2) use the cosine to find the base. 



PLANE TRIGONOMETRY 



347 



The sin, cos, and tan: Simple Inter-relations 

Introduce the notation a, b y and c to represent the number 
of units of length in the sides opposite the angles corre- 




spondingly named. Also show that since Z.A + Z.B = 90 
A = 90 B, and B = 90 A. Now tabulate: 

- = tanB, or b = a tanB; - = tanA, or a b tanA. 
a b 

- = sinB, or b = c sinB; - = sinA, or a = c sinA. 
c c 

_ = cosB, or a = c cosB; - = cosA, or b = c cosA. 
c c 

Hence, 

(1) Since ? = tanA, and - = tanB, /. tanA = 
b a 



(2) Since - = sinA = cosB, 



(3) Since - = cos A = sinB, 
c 



w l = b' 

c 

(5) Similarly, 



tanB 
sinA = cosB. 

cosA = sinB. 
sinA 



cos A 
sinB 



= tanA. 



tanB* 



348 



CRAFTSMANSHIP IN MATHEMATICS 



(6) Since B = 90 A, and sin A =- cosB, 

.*. sinA = cos(90 A). 

(7) Since B = 90 A, and cos A = sinB, 

/. cos A = sin (90 A). 

All these relations must be carefully committed to memory.* 
Note that the last two may be summed up in this way: the 
sine of an angle is the cosine of its complement. Explain the 
significance of co- in cosine. 

Some teachers prefer the words opposite and adjacent 
instead ^perpendicular and base, but experience suggests that 
for beginners the latter terms are preferable. The main 
thing is to adopt one form of words and stick to it. 

The secant, cosecant, and cotangent. These should be 
remembered as the reciprocals of the cos, sin, and tan, 
respectively. Give easy examples to show the appropriateness 
of the forms beginning with co. 

The ratios of common angles. The sin, cos, and tan of 
the common angles 30, 45, and 60 should be memorized 





Fig. 187 

as soon as the nature of the three functions is understood. 
Teach the boys to visualize the half square and the half 
equilateral triangle the obvious aids to memory. 

* Dp not despise some simple form of mnemonics when, with beginners, con- 
fusion is almost inevitable, as in the case of the three trigonometrical functions; e.g. 
remember 



(1) Tan = 

(2) Sin == 

(3) Cos = 



by the words Tanned Post Boy, 
by the words Sign, Please, Henry, 
by the words Costly Black Hat, 



or some other form of catchy words, 



PLANE TRIGONOMETRY 



349 





30 


45 


60 






1 


V3 


sin 


4 


V2 


T~ 




V3 


1 




cos 






i 




1 






tan 




1 


V3 




V3 







A little later, the table should be extended to and to 
90, and eventually to 180. When discussing the and 
90 values, draw a series of right-angled triangles, beginning 
with a very small acute angle A and very nearly 0, and ending 
with an angle A very nearly 90. A discussion of just one 
general figure, without reference to the actual values of 
particular cases, is, with beginners, almost profitless. Do 
not say that the tan of 90 is " infinity ", a term which is 
beyond the comprehension of beginners. Adopt some such 
non-committal form of words as " immeasurably great ". 



* 




Fig. 188 



The ratio for 15 is easily obtained from this figure, 
derived from fig. 187. ABC is an isosceles rt. Z.d A, sides 
1, 1, 2, angles 45, 45, 90. ABD is a half equil. A, sides 
1, 2, v% angles 30, 60, 90. Thus ^CAD = 15, and 



350 CRAFTSMANSHIP IN MATHEMATICS 

CD = (-\/3 1). From C, drop a perpendicular on AD. 

Since CED is a half equil. A, CE = CD = ^ 3 ~ 1 . 

pp A/3 _ 1 
Hence sin 15 -' ~ --- From this the other ratios 



of 15 are easily found, and then those of 75. 

For 18, fig. 113fl is the key. The small angles of a regular 
pentagram are 36, and hence the sine of half the angle is 

- - . Let the boys work this out for themselves; it is 
4 

a good exercise; the other ratios may be derived arithmetically, 
but the first (the sine) must be established geometrically. 
The derivation for multiples of 18 (36, 54, 72) is suitable 
work a year later. 

The following identities may readily be established 
geometrically. 

1. Sin 2 A + cos 2 A = 1. This is seen from a figure to 
be a direct application of Pythagoras. Let the derivatives 
also be noted: sin A = A/1 cos 2 A, cos A = A/1 sin 2 A. 

2. 1 + tan 2 A = sec 2 A. Here a hint is necessary to the 
boys to work " backwards ". We have to prove: 

J? j BC 2 AB 2 




i.e. 



AC 2 AC 2 ' 
AC 2 + BC 2 AB 2 



Fig. is 9 AC 2 AC 2 ' 



The boys now observe that the numerators form the simple 
Pythagoras relation. Hence they write out: 

AC 2 + BC 2 = AB 2 , (Pythag.) 

. AC 2 BC 2 AB 2 . , 

' 



/. 1 + tan 2 A = sec 2 A. Q.E.D. 

The obvious derivatives should follow. Give several easy 



PLANE TRIGONOMETRY 351 

examples to verify the rule that if any one trigonometrical 
ratio of an angle be given, the other ratios may all be cal- 
culated without reference to tables. But all fundamental 
relations must be established geometrically. Geometry must 
take precedence over algebra. 

Heights and Distances 

It is surprising what a great variety of problems, in 
three as well as in two dimensions, may be solved by means 
of the small amount of trigonometry already touched upon. 
Give plenty of such problems until the sin, cos, and tan 
are as familiar as the multiplication table, are, indeed, a 
part of the multiplication table. Insist all along that every 
problem on heights and distances is really a geometry problem 
with an arithmetical tail, but that the arithmetic is made 
easy for us because all the necessary multiplication sums 
have been worked out and the answers put into a book of 
tables, the multipliers having been given the rather fanciful 
names of sin, cos, tan, &c. In every problem we are con- 
cerned with a triangle; the length of one side is always 
given, and the multipliers in the book of tables enable us to 
find the other sides; to find the multipliers, we have to know 
the angles of the triangle. Four-figure tables of natural 
sines, cosines, and tangents, for whole degrees only, are 
enough for beginners. Let logs wait. Let the problems be 
easy and varied. Three-dimen- 
sional problems may be in- 
cluded quite soon, though at 
least a little solid geometry 

should have been done pre- W 

viously. 

When setting problems in- 
volving " bearings ", avoid, as ,. 
a rule, the old terms " north- Fig. 190 
west ", " south-east ", &c., and 
adopt the surveyor's plan, always placing N. or S. first, 



352 CRAFTSMANSHIP IN MATHEMATICS 

then so many degrees W. or E., thus N. 30 W., S. 60 E. 
the angle always being measured from the N. S. line. 

The drawing of figures for heights and distances. If a 
figure lies wholly in a horizontal plane, there is seldom much 
difficulty, especially if drawing to scale has been properly 
taught in the Junior Forms. Figures in a vertical plane are 
also readily drawn, though the angles of elevation and de- 
pression are sometimes confused by boys whose early practical 
geometry has not been properly taught. 

Consider this old problem: 
From a point P in a horizontal 
plane, an observer notes that a 
distant inaccessible tower subtends 
an angle of 30. He walks to 
Q, a distance of 100/J., towards 
the tower, and finds that the tower 
then subtends 50. Find the height 
Fig. 191 of the tower and the man's dis- 

tance from it. 

Explain how easy it is to work with tangents, as the figure 
readily shows. 

(1) RS = PS tan 30, i.e. x = (y + 100) tan 30. 

(2) RS = QS tan50, i.e. x=y tan50, 

.'. (y + 100) tan 30 - y tan 50. 

Hence y can be found, then x by substitution. The long 
succession of statements in some of the textbooks is un- 
necessary and merely serves to bewilder the boys. 

The problem is, of course, easy enough. It is only when 
the measured distance PQ is not in the same plane as PRS, 
i.e. is not directly towards the tower, that the boys are baffled, 
because of the difficulty of drawing a suitable figure in 3 
dimensions. 

We will deal with the three-dimensional figure difficulty 
in one or two problems: 

A wall 12 ft. high runs east and west. The sun bears 




PLANE TRIGONOMETRY 



353 



Calculate the breadth of the 



S. 60 W. at an elevation of 32. 
shadow of the wall on the ground. 

This is taken from one of the best of our textbooks, 
is, of course, very simple, yet S 

I have given it to several lots of 
boys, and the necessary figures 
have nearly always puzzled them. 
Had the sun been directly south, 
a stick placed vertically at O 
would have had its shadow cast 
on the ground in the direction 

ON (fig. 192). But as the sun was S. 60 W., the stick at O 
would have had its shadow cast on the ground in the direc- 



It 



W 



N 

Fig. 192 




W 



N 
Fig. 193 



tion OQ lt so that the shadow makes an angle of 30 
the vertical plane (wall) in EW (fig. 193). 

SUN 



with 




O 
Fig. 194 



But as the sun has an elevation of 32, the length of the 
shadow of the stick RO would be Q 2 O, Q 2 being the far end 
of the shadow on the ground (fig. 194). (During the day 



(B291) 



354 



CRAFTSMANSHIP IN MATHEMATICS 



this shadow would occupy a succession of positions, just as 
if it were pivoted on the stick, following the sun round.) 

We have to consider these two things, the direction and 
the length of the shadow in a three-dimensional figure. Let 
ABCD be the wall running east- west; it may be looked 
upon as a close set of palings, with one paling RO taking 
the place of the stick. Of course the shadow of the whole 
wall will be cast, but we will first consider the shadow of 
RO only. As the sun is at 32, the shadow must be cast 

rS\JH 




Fig. 195 



somewhere on the ground as a length OQ. That " somewhere " 
is given us by the sun's position (irrespective of its height) 
at a particular time in the day, viz. S. 60 W., i.e. OQ will 
make 30 with the east-west wall, or ZBOQ = 30. 

Now all the palings will cast shadows parallel to OQ, 
and thus we shall have a belt of shadow, on the ground 
BMNC, of a breadth equal to the perpendicular QP to the 
wall. Thus we have to find the length of PQ. 

To find the length of PQ we may solve the APQO in the 
H.P. In the ARQO (vertical plane), RO = 12'; QO = RQ 
cot32 = RO tan58 = 12' x 1-6 = 19-2'. In the APQO, 
PQ = QO sin30 = 19-2' X -5 = 9-6'. 



PLANE TRIGONOMETRY 



355 



For beginners a model is far better than a sketch; then 
the angles do not mislead. Even a book held upright on the 
desk to represent a vertical plane, and then a pencil placed 
in position to represent a line in an oblique plane, will help 
the eye greatly. But some long hat-pins stuck vertically 
into a board, with pieces of cotton tied round under the 
heads (a snick made with the laboratory file will help to 
secure the cotton), stretched and held fast by a twist under 
the head of a drawing-pin, will enable the boys to make in 
a minute or two a model of almost any figure that may be 
required. 

Here is another problem and the provided figure from the 
same excellent textbook. The figure has puzzled several 
lots of boys. 

" A hillside is a plane sloping at 27 to the horizontal. A 
straight track runs up the hill at an angle of 34 with a line 
of greatest slope. What angle does the track make with the 
horizontal?" 

" AB is the line of intersection of the hillside and a 
horizontal plane ABC. AF, BE are lines of greatest slope 




Fig. 196 



meeting a horizontal at F, E. Let the track AD cut EF at 
D. Draw DN, EC perpendicular to the H.P., ABC. Then 
AN is the projection of AD on ABC. It is required to 
find ZDAN = 0, say." Then follows the solution, simple 
enough, of course, from considerations of the 3 it. Zd As 
ECB, AND, AFD, the first two in V.P.s, the last in an oblique 



356 



CRAFTSMANSHIP IN MATHEMATICS 



plane. A model with 3 hat-pins at FM, DN, and EC, and 
drawing-pins at A, B, C, M, N, and connecting threads, 

would make the whole 
thing clear at once. 
Otherwise a few shading 
lines might be added, as 
in fig. 197; the 3 planes 
are then shown clearly. 

Here is a simple 
problem from another 
book, the figure for which 
has often given beginners 

trouble. The extremity of the shadow of a flag-staff FG, 6' 
high, standing on the top of a square pyramid 34' high, just 
reaches the side of the base and is distant 56' and 8' respectively 
from the extremities of that side. Find the sun's altitude. 




Fig. 197 





Fig. 198 

FK = FG + GK = 6' + 34' = 40'. We have to find 
the Z.FMK in the rt. Zd A FMK. In this A we know 
FK; and we can find MK by Pythagoras from A KMN 
in the plan (second figure): 



KM = A/32 2 + 24 2 = 8 x 5 = 40. 
TanFMK = $g = 1; /. ^FMK = 45. 



PLANE TRIGONOMETRY 



357 



In practical problems, boys are constantly blundering 
over compass bearings. Impress on the class that the difference 
between the bearings of two distant objects is the angle made 
by the two lines, drawn in the Jf/.P., from the observer to 
the objects. If the objects are above the H.P., the difference 
between the bearings is still an angle on the H.P., viz. the 
angle between the two vertical planes drawn through the 
observer and each of the objects. An observer is at S in a 
H.P., his south-north line being SN. PQ and RT are two 
vertical poles. He takes the bearings of the two poles and 




Fig. 199 

finds that their horizontal angles are respectively N. 50 W. 
and N. 55 E., i.e. the difference between their bearings is 
105. 

Now suppose he could not see the bottom of the poles, 
because of an intervening hill. The observer would have 
to point his telescope at the pole-tops P and R, and he could 
then, if he wished, take the angles of elevation. But his pur- 
pose now is to take the difference between the bearings, and 
he would therefore observe where each vertical plane con- 
taining the tilted telescope cut the horizontal plane. The 
angle to be measured ( ZQST) would be exactly the same 
as before. Impress on the boys that the observer could not 
measure the angle PSR in the oblique plane; his theodolite 
does not permit of that. And even if he could, the angle 
would not represent the angle between the bearings. 

Here is an illustrative problem. Find the distance between 
the tops of the spires of two distant inaccessible churches. (It 
would be taken rather later in the course.) 



358 



CRAFTSMANSHIP IN MATHEMATICS 



Measure off a base line AB in a suitable position, and from 
each end take the bearings of both spires, P and Q, draw 




ELEVATION 
(V P *o.tl> spines) 



a ground plan, and mark in the Zs a, j8, y, 8. The 
f may be calculated if wanted. 



and 



In the APAB, AB and the Zs are known; 

.*. PA and PB can be calculated. 
In the AQAB, AB and the Zs are known; 

.'. BO can be calculated. 
In the APBQ, PB and BQ are calculated, and ZS known, 

/. PQ can be calculated. 

Now examine the perspective sketch, with the 2 spires P'P 
and Q'Q in position. We have to find P'Q'. We know AP, 
BQ, PQ. Measure the /.s of elevation p and a\ P'P = AP 
tan/), Q'Q = BQ tancr; hence P'P and Q'Q are known. 
Hence in the elevation, everything is known except P'Q', 
and this is easily calculated by Pythagoras. 

Make the boys do this practically. Any two distant tall 
objects will do. 



PLANE TRIGONOMETRY 369 



The Obtuse Angle 

Angles up to 180 should be considered at an early stage, 
but, before angles greater than 180 are considered, substantial 
progress on the practical side of the subject is desirable. 

Remind the boys that the rotating arm of the angle, 
regarded as the hypotenuse of a rt. Zd A, may be carried 
round from 90 to 180, the pivot being the point of inter- 
section of the co-ordinate axes. Refer to the work on graphs, 
and the rule of signs for the second quadrant; all x values 
measured to the left of the origin are regarded as negative. 
If, for instance, we consider 
the triangle BOC in the 
second quadrant, the hypo- 
tenuse and perpendicular are 
positive as in the first quad- 
rant, but the base OC is 
negative. Proof? There is 
none. It is merely an accepted Fig. 201 

convention. 

Suppose, then, we have an angle greater than 90, say 
145. How am I to find the value of its sine, cosine, and 
tangent? 

Exactly as before. From any point on the rotating 
arm OB, drop a perpendicular to the fixed arm OA (pro- 
duced backwards, because necessary), and take the ratios 
in the same way as we did for acute angles. But remember 
the signs. For this angle 145, the hypotenuse is OB, the 
base OC, the perpendicular BC. 

TT IJfO BC 1JKO OC 1 AtaO BC 

Hence: sin 145 = ; cos 145 = , tan 145 = 




OB' ~" w OB~ 

It may not look as if the perpendicular BC concerned the 
angle AOB (145), but how else could a perpendicular for 
145 be drawn? 

Now consider an acute angle equal to the angle BOC 
in fig. 201. Evidently it is (180 - 145) or 35. Fig. 202 



360 



CRAFTSMANSHIP IN MATHEMATICS 



shows OB' equal to OB in fig. 201. Hence the triangle 
B'OC' has sides of exactly the same length as the triangle 

TVP' OP' ' 

BOC. SinS5' = fL; cos35 = ; tan35 = 




Fig. 202 

Comparing the ratios for 145 and 35, we see that: 

sin35 = sinl45, 
cos35 = _cos!45, 
tan 35- -tan 145. 

The same thing must apply to any pair of angles whose sum 
is 180. Thus we may say, 

sinA = sin(180 - A), 
cosA = -cos(180 A), 
tanA = -tan(180 - A). 

If the above demonstration is attempted from a single 
figure (as it might well be from fig. 201), slower boys will 
inevitably be confused. 

It is an excellent plan to make boys in Upper Sets express 
their ratios in terms of co-ordinates, i.e. to call the rotating 
arm r, and its extremity P (#, y). 

Give plenty of oral practice in the obtuse angle relations, 
e.g. tanlOO = tan(180 - 100) = -tan80 = -5-67. 



PLANE TRIGONOMETRY 361 



The General Triangle and its Subsidiary 
Problems 

Before proceeding to the solution of triangles, revise 
carefully the geometry of congruent triangles, and note 
what various sets of data are necessary and sufficient for 
copying a triangle. A triangle is determined uniquely if we 
are given (1) the 3 sides, (2) 2 sides and the included angle, 
(3) 1 side and 2 angles. If we are given 2 sides and the angle 
opposite one of them, there may be 2 solutions, or 1 solution, 
or no solution. 

A triangle cannot be determined unless the data include 
at least one side. 

Thus the necessary data include 3 elements, at least 
one of them being a length. 

All the formulae in this section must be established 
geometrically. As geometrical exercises they are all first- 
rate. 

1. In any triangle ABC, a = b cosC + c cosB. Show 
that this relation holds good for both acute-angled and obtuse- 
angled triangles. It is simply a question of dropping a per- 
pendicular and considering separately the two resulting 
right-angled triangles. 

Do not forget the sister expressions in this and sub- 
sequent formulae. The one thing to keep in mind is the cyclic 
order of the letters A, B, C, and a, A, c. For instance, the 
above identity may be written, 

b = c cos A + a cosC, 
or, c = a cosB + b cos A. 

2. The sine formula. In any triangle ABC, 



sin A sinB 
As before, show that the relation holds good for 



sinC 
obtuse-angled as well as for acute-angled triangles. 



362 CRAFTSMANSHIP IN MATHEMATICS 

The " ambiguous case " should receive special attention. 
Link up the work with the closely analogous case in geometry. 
In fact the problems are the same. Readily understood as 
they generally are, they are often half forgotten. They must 
be regarded as sufficiently tedious and troublesome as to 
merit special and repeated attention. 

3. The cosine formula. In any triangle ABC, c* = a 2 -f ^ 2 
~-2ab cosC. Again be careful to consider both acute-angled 
and obtuse-angled triangles. Link up carefully with Py- 
thagoras and its extensions (Euclid, I, 47; II, 12, 13). The 
solution is straightforward and seldom gives trouble. 

When solving triangles, use sine or cosine formula? 



If given (1) 3 sides, 

or (2) 2 sides and in-, 
eluded angle, 



use cosine formula for first 
operation; then continue 
with the quicker sine for- 
mula, using it to find the 
smaller of the two remain- 
ing angles. 

If given (3) 2 angles and 1 side, } use sine 

or (4) 2 sides and a not-included angle,) formula. 

N.B. (1) If given 3 sides, find the smallest angle first. 

(2) If the given triangle is isosceles, use neither 
formula, but drop a perpendicular to the 

base. 

* 

4. The tangent formula. In any triangle ABC (where 
b>c), 

tanj-(B C) _ b c 
tan|(B + C) ~~ b + c 

This is a useful alternative, more suitable for log calculations, 
when 2 sides and the included angle are given. The cos 
formula is often cumbrous in application, not being suitable 
for log calculation. 



PLANE TRIGONOMETRY 363 

The boys must learn to establish the formula geometrically, 
from first principles, and not derive it from other trigono- 
metrical formulae. But for beginners it is generally puzzling. 
Begin by giving them a particular case to which they may 
apply the formula. Let the sides of a triangle be, say, 11, 
13, 16, and let the boys work out the angles from their cos 




B a 16 O 

Fig. 203 

and sine rules, using four-figure tables. The angles shown 
in the figure are, to the nearest minute, 

tan|(B - C) = tan |(53 47' -43 3') 
tanJ(B + C) tan|(53 47' + 43 3') 

tan522 ; -094 1 



Again: 



tan 48 25' 1-127 12* 
b-c 13-11 1 



b + c 13 + 11 12 



Thus they see that, at least in this particular case, the theorem 
holds good. Working out a particular case in this way, they 
grasp the fact that tan|(B C) is, after all, just the tan of 
a simple angle. So with |(B + C). 

The problem now is to devise a figure which shall 
actually show these angles (B C) and f (B + C); also 
the sum (b + c) and the difference (b c) of the sides. 

There are two subsidiary points to note first. 



364 CRAFTSMANSHIP IN MATHEMATICS 

(1) In any triangle, 

since A + B + C = 180, 



or 

Again: 

or 
Similarly: 



+ 



_ 

-90 --. 



T' 



B f C 



B-C 



22 

t_A _ A = C 



Give plenty of oral work on these points, with blackboard 
figures to illustrate. 

(2) How have we been able in geometry to show the 
sum and difference of two sides of a triangle? 





Fig. 204 



The sum of b and c may be shown by swinging round AB 
on A to AE, so that AE = AB; hence CE = (b + r); the 
difference may be shown by cutting from AC a part AD equal 



PLANE TRIGONOMETRY 



365 



to AB; thus DC = AC AB = (b c). The same figure 
shows (B C). For (fig. ii), since m n + jf>, .". m' = n + />> 
/. m' + p = n + 2p, or B =- C + 2p; ;. == J(B C). 

(i) Now we may draw the required figure. 

With centre A and radius AB, describe the circle EBD, 

and produce CA to E. Evidently EC = (b + c), DC = (b - c). 

Join EB. Z.E (at circf.) = ^A (at centre). We know that 

/.DEC |(B C), but there seems to be no obviously 




Fig. 205 



simple way of using it. But if we draw CF parallel to BD, 
to meet EB produced in F, Z.BCF = ZDBC = (B C). 

Again, in the right-angled triangle EFC, since /.E = A, 
ZECF = 1(B + C). 

Thus we have the two angles and the two lengths for the 
tan formula: 

BF 

tan^(B-C) = tanBCF = FC = BF = DC _ b c Q E D 
tanJ(B + C)~~tanECF EF EF EC b + c * * 

FC 

(ii) A boy might very well ask if we could use the figures 



366 CRAFTSMANSHIP IN MATHEMATICS 

made by drawing the circle with radius AC instead of AB. 
Exterior ZA at centre = B + C; /. ZD at circum- 
ference = |( B + C). 




Fig. 206 

Also /.BCD = ZB LD (ext. L property) = ZB 
- |(B + C) - KB - C). 

We may take the tan of the last angle by dropping the 
BF. 

BF 

tan(B~C) _ tanBCF FC _, FD _ BD _ b-c QED 
tan |(B + C) tanBDF BF FC BE b + c ' " 

FD 

There is no essential difference between the two proofs. 

(iii) Or a boy might ask if we could not derive the angle 
+ C) from the JA obtained by actually bisecting A. 



PLANE TRIGONOMETRY 



367 



Let AD be the bisector, and let CD meet it at right angles. 
Draw BF perpendicular to CD produced, and BE perpen- 
dicular to AD. 

Evidently ZBCF 
= KB C), and 
Z.AEE = ZACD = 
KB + C). 

The figure does 
not give us a length 
AC + AB (= b + c), 
or a length AC AB 
(= b c). But we 

can project AB and Fig 20? 

AC on to FC; FD 

(= BE) is the projection of AB, and DC is the projection 
of AC, and so we may obtain what we want in this way: 

(1) AD = b sinKB + C); AE == c sin|(B + C); 
.-. BF = AD - AE 




Vi(B-c) 



(2) DC = b cosKB + C); FD = c cos(B + C): 

/. FC = DC + FD 

= (b + c) cos|(B + C). 

. BF _ (b - c) sinj(B + C) _ b - c 
(6) FC ~ (4 + c) cos KB + C) V+c a i( 



But 



BF 
= tani(B 



C); 



. 
b 



tan |(B + C) = tan KB - C), 

b c = tan J(B C) 
b + c ~~ tan KB + Cj" 



or 



This last method is not quite so simple as the first, but it 
appeals to A Sets to whom alone (perhaps) it should be given, 



368 CRAFTSMANSHIP IN MATHEMATICS 

5. Other formulae that should be worked out geometrically: 

(i) Cos2a = 2cos 2 a 1 = 1 2 sin 2 a. 
(ii) Sin2a = 2 sin a cos a. 

02 I 2 _ C 2 

(iii) CosC -- -- - ---- (and thence, algebraically, 



. -- 

the half-angle formula, sin|A == y ~ *) (* ~ c ) 

o \ be * 

&c.). 

(iv) Area = \bc sin A; &c. 
(v) Circles of a triangle: circumscribed, inscribed, 

escribed. 
(vi) Medians, angle bisectors, pedal triangle, ortho- 

centre, &c. 

The geometry of these basic formulae is the important thing. 
The derivatives may be obtained algebraically. 

Angles up to 360. The Four Quadrants 

It is best to begin by showing the boys how surveyors 
in their work often find it an advantage to consider angles 
up to 360. We have therefore to decide how the ratios of 
angles between 180 and 360 can be expressed. Thus we 
have to consider the 3rd and 4th quadrants. 

Remind the boys that there are no proofs of our conven- 
tions concerning the signs in the four quadrants. The con- 
ventions are just a matter of convenience, arrived at by 
general consent, and consistent with one another. It is this 
consistency which is the important thing. The boys must 
be drilled in the quadrant signs until the last shred of doubt 
disappears. 



44 Plus: right and above," 
44 Minus: left and below." 

Fig. 308 



PLANE TRIGONOMETRY 



369 



Other important memos. 

1. The fixed arm of the angle is always in the 3 o'clock 
position. 

2. The rotating arm of the angle always moves counter- 
clockwise. 

3. Never take a short cut by moving clockwise. 

From any point in the rotating arm we may drop a per- 
pendi cular PM on the abscissa, form a right-angled triangle, 





M. 



(0 



(ii) 





Fig. 209 



and so take any ratio of any angle in any quadrant. Taking 
e.g., the tangent, we have: 



OM 2 
tanXOP, = d^ = 



tanXOP = 
ta 4 



~ 
+OM 4 



OM 2 ' 



OM' 



-. 
OM 4 



(1291) 



25 



370 



CRAFTSMANSHIP IN MATHEMATICS 



Beginners are often puzzled about the re-entrant angles in 
the 3rd and 4th quadrants. Make them understand that if 
they take the smaller angles in these quadrants, they have 
taken a clockwise rotation of the moving arm, and this is 
not allowable. (Postpone the consideration of negative rota- 
tions until the main principle is grasped thoroughly.) 

As already suggested, an alternative plan is to call the 
length of the rotating arm r, and to call the point P which 
we fix in it (x, y), x and y being the co-ordinates of the point. 
But if the boys are at first well drilled in the use of the terms 
hypotenuse, base, and perpendicular, these terms will probably 
continue to be used, at least mentally. In A Sets, the co- 
ordinate notation is preferable: its advantages are obvious. 

Make the boys memorize the following scheme: it merely 
amplifies what was said on a previous page. 





i. 


2. 


3. 


4. 


Sin . . 


+ 


+ 








Cos .. 


-f- 








+ 


Tan . . 


+ 





+ 






" Sin, cos, and tan are + i n the 1st quadrant, and each is + in 
one other, viz. sin in 2nd, tan in 3rd, cos in 4th."* 

Give plenty of oral work on the ratios of angles in all 
four quadrants. Boys should recognize the landmarks 90, 
180, 270, 360, and know at once in which quadrant a 
given angle occurs. 

Beginners are often caught: they take the complement 
of the angle instead of the angle itself. Point out again and 
again that whatever angle we may have in the first quadrant 
there must be angles with exactly the same numerical ratio 
in the other three quadrants. The four resulting triangles 



* One or two schools use this mnemonic: 
s * |- all 
T I 



" positively all silver tea cups ". 



PLANE TRIGONOMETRY 



371 



formed by dropping a perpendicular from the same point 
P on the rotating arm must be congruent. 







/ \ 


N^ 






1 


+ 1-28 +1-28 




> 





"*" 


X 

as: 




232 



(iii) 




Fig. 210 



Note the 4 angles: 52; 180 - 52 = 128; 180 + 52 
= 232; 360 - 52 = 308. 

1*28 
Note also the 4 tangents:* tan52 = + - ; tan!28 



= - ; tan232 = 



; tan308 = - 



The angle in the second quadrant is obtained by sub- 
tracting 52 from 180. - 

The angle in the third quadrant is obtained by adding 
52 to 180. 

The angle in the fourth quadrant is obtained by sub- 
tracting 52 from 360. 



Quite by chance the angle in the second quadrant (128) has the appearance of 
being 100 times the value of the tangent (1-28). 



372 



CRAFTSMANSHIP IN MATHEMATICS 



The four angles do not form an arithmetical progression, 
and they cannot do so unless the angle in the first quadrant 
is 45. 

Any such group of 4 angles forms a symmetrical figure: 




Whenever we take a trigonometrical ratio of an angle 
from the tables, the angle is one belonging to the first quad- 
rant. But there are three other angles having the same 
numerical value. If a is the angle in the first quadrant, the 
other three are 180 - a, 180 + a, 360 a. But each 
ratio in each quadrant has its own signs as we have already 
seen. 

To evaluate the ratios of angles greater than 90, we may 
remember the formula, 180 a, though this is really 
intended to include angles greater than 360. Let the boys 
make up this general formula from an examination of a 
number of particular cases. 



First Notions of Periodicity 

The boys are already familiar with the notion that the 
rotating arm of the angle may proceed beyond one revolution; 
the movement of the pedal of an ordinary bicycle serves to 
convey the notion of angles of w360 or w360 + a. Show 
clearly that the ratios of any angle a are exactly the same as 
those of any angle that differs from a by any number of 



PLANE TRIGONOMETRY 



373 



complete revolutions. Thus, sin(w360 + a) sin a, where 
n is any integer; so with all the ratios. For example, 

cos 700 = cos (720 - 20) - cos (2 . 360 20) 
= cos 20 = cos 20. 

Give ample oral practice to emphasize the fact that the addition 
or subtraction of any multiple of 360 does not alter the value 
of any ratio of an angle. The general rule may be expressed: 
" If a is an angle, any ratio of %nir i a is numerically equal 
to the same ratio of a ". The sign to be attached depends 
on the quadrant. (The radian notation should be familiar 
by this time.) 

For purposes of illustrating continuous functions, graphs 
may be obtained, with sufficient accuracy, by 30 and 60 




360 




360 




Fig. 212 

parallels and perpendiculars as in fig. 183. The two inter- 
mediate points in each quadrant are enough to determine 
the curve fairly readily. The boys should be able to sketch 
the curves rapidly and should become thoroughly familiar 
with them. They should note that if the graph of cos a be 
moved 90 units along the x axis, it coincides with that of 
sin a, and that this is equivalent to saying that sin (a + 90) 
= cos a.* Superpose the cos graph on the sine graph and 
discuss the intersecting points and the ratios of the angles 
indicated by those points. Draw a continuous sine graph up 
to 5?r or GTT. Select some first quadrant angle, say 40, 
raise a perpendicular to cut the graph, and through the 
point of intersection run a parallel to the x axis and another 
the same distance below the axis. Discuss and compare the 



Slower boys will confuse 90 -f A with 180 A. 
necessarily different unless A = 45. 



Show that they are 



374 



CRAFTSMANSHIP IN MATHEMATICS 



sines of all the angles indicated by the successive points 
of intersection. Show clearly that there is a period of 27r, 
and that sin# may therefore suitably be called a periodic 
function of x. So with cos x. Tan x is likewise a periodic 



7 



~3ir 



\ 



lig. 213 



function, but with a period of TT (not 2??); show how this 
may be inferred from the parallel tan curves. 



Compound Angles 

1. Sin(A + B) = sinA cosB + cosA sinB. 

2. Cos(A + B) = cosA cosB sinA sinB. 

3. Sin(A B) = sinA cosB cos A sinB. 

4. Cos(A B) cos A cosB + sinA sinB. 

Beginners naturally think that sin 50 = sin 20 + sin 30, 
that cos 70 = cos80 cos 10. Give a few examples, with 
free reference to the four-figure tables, to show that this is 
not so. 

Of the four identities named above, at least one should 
be proved geometrically and mastered thoroughly. The 
neatest method is the projection method, and with A Sets 
the general case can readily be proved by this method. With 
B Sets and certainly with C Sets the problem is best con- 
sidered merely from the point of view of positive acute 
angles. All the books give the solution, but the boys should 
be taught to analyse the conditions of the problem, not 
merely to follow out a book solution. 

The following sequence of arguments is suitable for teach- 
ing purposes. 



PLANE TRIGONOMETRY 



375 



Let OX rotate through ZA to OC, then through ZB 
to OD; in its complete journey to OD it has rotated through 
the complete L (A + B). We have to prove that sin (A + B) 
= sin A cos B + cos A sin B, and in connexion with the 
three angles this means the consideration of five ratios, viz. 




Fig. 214 



the sines of A, B, and A + B, and the cosines of A and B. 
We will try to show all these in one figure. 

Evidently we require three perpendiculars, since there 
are three angles. 

(1) From any point P in OD, drop the JL PN to OX. 
The sine of L(A + B) may be considered from the rt. Zd A 
PON. 

(2) From P, drop a -L PQ on OC. The sine and the cos of 
ZB may be considered from the rt. Zd A POQ. 

(3) From Q, drop a -L QR on OX. The sine and cos of 
ZA may be considered from the rt. Zd A QOR. 

When we have to prove that a simple expression is equal 
to a more complex expression, it is a good general rule to 
begin with the latter, try to simplify it, and get back to the 
former. Thus we may begin: 

sin A cosB -f- cos A sinB 

= cm QO OR PQ 

QO ' OP + QO ' OP' 
But how are we to proceed now? True the OQ's seem to 



376 



CRAFTSMANSHIP IN MATHEMATICS 



cancel out in the left-hand term, but we do not seem to be 
able to simplify any further. 

Since a circle will go round ONQP (on OP as diameter), 
Z.MPQ = ZA. If then we draw QM PN, we have a 




N R X 



Fig. 215 



APMQ similar to AQRO. Thus, as far as ratios are con- 
cerned we may consider APMQ instead of AQRO. Now 
let us try simplification again: 

sin A cosB -f cos A sinB 



Q0 , PM PQ 
'OP PQ'OP 



QR . PM PN . /A , m 

c= - + - = - = sin (A + B). 
OP OP OP V J 

We may now set our proof as an examiner would expect to 
see it. 

QR PM 




_ QR OQ PM PQ 
"~OP'OQ OP 'PQ 

(each term multiplied by 1) 

= QR OQ PM PQ 
OQ ' OP + PQ ' OP 

= sinA sinB + cosA sinB. Q.E.D. 



PLANE TRIGONOMETRY 



377 



The three analogues now follow on simply. All four identities 
should be verified by a few particular cases (4-figure logs will 
do), e.g. 

(sin55 cos 25 + cos 55 sin 25, 



sin 80 = 



IsinlO cos 70 + cos 70 sin 10. 




Fig. 216 



Some teachers prefer this proof instead: 

Let the acute Z.s A and B 
be the Zs of a A ABC. Draw 
a circle round the A, and the 
radii OA, OB, OC. Evidently 
ZAOC - 2B, ZBOC = 2A. 
If J-s from the centre be 
drawn, they bisect the sides 
of the A . Hence, AB = d sin 
(A + B), AC -rf sinB, CB 
= rfsinA; also AB = AC 
cos A + CB cosB. By equat- 
ing the first and last of 
these, and substituting from the 2nd and 3rd: 

d sin(A + B) = AC cos A + CB cosB 

= d sinB cos A + d sin A cosB, 
or sin(A + B) sinA cosB + cosA sinB. 

This proof does not seem to appeal to boys so readily as the 
former does. 

We now come to the general case. B and C Sets find it 
difficult, and as a rule it should be given to A Sets only. 
Of the various methods of proof the two following are the 
simplest for teaching purposes. 

1. The Projection Method. This method is productive 
of mistakes unless the boys have mastered the elementary 
principles of projection. 

Give the class one or two preliminary exercises of the 
following kind: 



378 



CRAFTSMANSHIP IN MATHEMATICS 




The angle A of the regular pentagon ABODE touches the 

X axis, with which AB makes 
an angle of 12. Find (1) the 
horizontal distance of the ver- 
tex D from A, and (2) the 
height of D above the X axis. 
(1) Horizontal distance of 
D from A = projection of 
AB + projection of BC + 
projection of CD. Remem- 
ber that projection means 
projection with proper sign 
attached, and we must take 
the Zs which AB, BC, CD 
make with the + direction of OX. Take AB as unity. 

/. Distance AH - AB cos 12 + BC cos 84 + CD cos 156 
= -9781" + -1045" -9135" 
= 1691". 

(2) Height of D above OX. 

Height HD - OM + MN + NP 

= AB sin!2 + BC sin84 + CD sin!56 
= -2079" + -994:5" + 4067" 
= 1-6091". 

Now we come to the identity sin(A + B) = sin A cosB + cos A 
sinB. Let the Zs be the same as in fig. 214. From any 




Fig. 218 



PLANE TRIGONOMETRY 



379 



point P in OD, draw PQ -L OC. Project the three sides of 
the APOQ on the Y axis. 

OM = ON + NM, 

.'. projection of OP = sum of projections of OQ and QP. 

/. OP sinXOP - OQ sinXOC + QP cosXOC. 
.'. OP sin(A + B) OP cosB sinA + OP sinB cosA, 

(OQ = OP cosB, QP = OP sinB) 

i.e. sin (A + B) = sin A cosB + sinB cos A. 

In a similar manner, by projecting the three sides on the 
X axis, we may prove that 

cos (A + B) = cos A cosB sin A sinB. 

Note that the method is perfectly general, being applicable 
to any angles. 

2. The Cosine Rule Method. This is based on the 

rules (1) that cos A = + c ~~ a _ and ^ 2 ) that if P and Q 

2bc 

are the two co-ordinate points (x l9 jy t ), (x 2 , j> 2 ), then PQ 2 

The identity usually considered is cos (A B) = cos A 
cosB + sin A sinB, the 
others being treated as 
derivatives. 

Whatever two angles 
are given, the initial line 
for each is the positive 
direction of the X axis. 
Note that we are taking 
the difference between 
two angles, not their 
sum. No matter what (CosA,s^A) 
two angles are taken, Fig. 219 

cosPOQ = cos(A B). 

The simplest way is to take a circle of unit radius, and 



Q(COS 




380 CRAFTSMANSHIP IN MATHEMATICS 

to let the co-ordinates of P be (cos A, sin A) and of Q, (cosB, 
sinB). Note that OP = OQ = 1; hence the denominators 
of the cosine ratios need not be written. 

Cos (A- B) 

= cosPOQ 

= Qp2 + Q 2 - PQ 2 

2OP.OQ 
_ 2 - PQ 2 

2 

2 - ((cos A - cosB) 2 + (sinA-sinB) 2 } 

. 



2cosAcosB + sin 2 A + sin 2 B 2sinAsinB} 

2 

2 {2 (2 cos A cos B + 2 sin A sinB) } 

2 
= cos A cosB + sin A sinB. 

There does not seem to be much to choose between this 
method and the project! ve method. To able boys both 
methods appeal. To boys of poor mathematical ability, 
both methods are equally hateful. 

One or other of the four identities, preferably the one 
proved by the projective method, should be regarded as 
basic, and the others should be treated as derivatives. 

Other necessary derivatives are: 

(1) 2 sin A cos B = sin (A + B) + sin (A B), and its 
three analogues. 

C + D C D 

(2) sinC + sinD = 2 sin cos - , and its three 

analogues. 

(3) tan(A + B) = tanA + tanB 
v ' v i y 



/A\ OA 

(4) tan 2 A = - . 
v ' 1 tan 2 A 

(5) sin3A = 3 sin A 4 sin 3 A. 

(6) cosSA = 3 cosA + 4 cos 8 A. 



PLANE TRIGONOMETRY 381 

All the formulae should be learnt off. If mnemonics can be 
devised, they will help the lame ducks much. 

Let the boys verify all the formulae established, by means 
of a few simple exercises. Use four-figure logs for this purpose, 
and so cover a good deal of ground in a short time. For 
instance, show that 

2 sin 50 cos 24 = sin (50 + 24) + sin (50 24) 

= sin74 + sin26, 
(2 X -7660 X -9135) = (-9613 + -4384), &c. 

Books to consult: 

1. Trigonometry, Siddons and Hughes. 

2. Elementary Trigonometry, Durell and Wright. 

3. Advanced Trigonometry, Durell and Robson. 

4. The Teaching of Algebra, Nunn. 

5. Elementary Trigonometry, Heath. 

6. Trigonometry, Lachlan and Fletcher. 

7. A Treatise on Plane Trigonometry, Hobson. 



CHAPTER XXVII 

Spherical Trigonometry 

Spherical trigonometry enters into the work of the map- 
maker, the navigator, and the astronomer; also into the 
work of the surveyor if that work extends over larger areas, 
as in the case of the Ordnance Survey. But for an under- 
standing of the essentials of surveying, map-making, navigation, 
and astronomy, little more than the A, B, C of spherical 
trigonometry is required, and all this can be included in a 
very few lessons. The elementary geometry of the sphere 
should already have been done. 

The following are the chief points for inclusion in the 
necessary elementary course. (Many of the difficulties can 



382 CRAFTSMANSHIP IN MATHEMATICS 

be elucidated by the use of simple illustrations. The orange, 
with its natural sections, is very useful. Well-shaped apples 
lend themselves to the making of useful sections. A slated 
sphere, mounted, should always be available). 

1. Great and small circles. 

2. Shortest distance that can be traced between two 
points on the surface of a sphere the arc of the great circle 
passing through them. (A simple experimental verification 
is good enough for beginners.) 

A suitable argument: If a string be stretched between 
two points on the surface of a sphere, it will evidently be 
the shortest distance that can be traced on the surface between 
the points, since, by pulling the ends of the string, its length 
between the points will be shortened as much as the surface 
will permit. Any part of the stretched string, being acted 
on by two terminal tensions, and by the reaction of the 
surface which is everywhere normal to it, must lie in a plane 
containing the normal to the surface. Hence the plane of 
the string contains the normals to the surface at all points 
of its length, i.e. the string lies in a great circle. (Sixth 
Form boys ought to appreciate such an argument.) 

3. Axes; pole and polar. 

4. Primary and secondary circles. 

5. The angle between two great circles is measured by: 

(i) the angle between their planes, 

(ii) the arc intercepted by them on the great circle 

to which they are secondaries, 
(iii) the angular distances between their poles. 

6. The spherical triangle that portion of the surface of a 
sphere bounded by the arcs of three great circles. Parts: 
3 sides and 3 angles. 

7. Since 3 great circles intersect one another to form 8 
triangles, that particular triangle is selected which has 2, or 
if possible 3, sides each less than a quadrant. 

Cut an orange or an apple into two equal parts; 



SPHERICAL TRIGONOMETRY 383 

hold the two parts together, and cut again into two 
equal parts, this time by a plane oblique to the first; 
hold the four parts together, and cut still again into 
two equal parts, by a plane oblique to both of the 
other planes. 

8. The analogy between theorems in plane and spherical 
trigonometry, e.g. any two sides of a triangle are together 
greater than the third. 

9. Polar triangles, i.e. triangles so related that the vertices 
of the one are the poles of the sides of the other. 

10. Angular limits of the sides and angles of a spherical 
triangle. 

11. Fundamental formulae: 

(i) Any spherical triangle: 

. cos a cosi cose , , 7 x 

cos A = : - . (and analogues). 

sm0 sine 

(ii) Right-angled triangles: 

sin A sin a/sine; 
cosA = tani/tanc; 
tan A = tanfl/sini. 

(iii) Sine rule: 

sin A sinB sinC 2n 



sin a sin b sine sin a sin b sine* 

All the proofs are simple. The only trouble is in the drawing 
of suitable figures. 

12. The Latitude problem. This is perhaps the most 
important of the elementary problems of the sphere. 

The navigator's " dead reckoning " depends on his know- 
ledge of two things: (1) his course (direction), (2) the distance 
run (determined by log). He has to resolve his distance- 
course into separate mileage components of northing and 
southing, easting and westing. Then he has to convert his 



384 CRAFTSMANSHIP IN MATHEMATICS 

northing and southing mileage into degrees and minutes of 
latitude, his easting and westing into degrees and minutes of 
longitude. 

There is no difficulty with the former. The meridians 
of longitude are all great circles. When we know the length 
of the circumference of these circles, a simple calculation 
will give the change of latitude produced by a given northing 
and southing. (Polar circumference 24,856 miles; therefore 
length of degree of latitude = 69 miles; - 6 1 6 of 69 miles 
= nautical or sea mile = 6080 ft. Thus 60 sea miles 1 
degree of latitude, and 1 sea mile = 1 minute of latitude. 
" Knots " = sea miles per hour.) 

But parallels of latitude are small circles decreasing from 
the equator to the poles. Only along the equator itself does 
1 sea mile imply 1 degree of longitude. We have to discover 
a law which the length of a degree of 
longitude follows. 

This law does not show a length 
proportional to the distance from the 
pole. The greatest distance between 
two meridians is not halved at 45, 
but at 60. Why has the parallel of 
60 half the circumference of the 
equator? 

Fig. 220 CE = radius; A = point in lat. 

60. Let figure rotate on PP'. The 

circle will trace out the surface of the globe, E will trace 
out the equator, and A the parallel of 60 of which AB 
is the radius. 

CE - CA = R (say). 
Then AB = AC sin ACB, 
= Rcos60, 
= *R. 

Since AB = |R, circf. of the 60 parallel = \ length of 
equator, .*. the length of a degree in 60 lat. is half the length 
of a degree along the equator. 

Thus a voyage of a given number of sea-miles along the 




SPHERICAL TRIGONOMETRY 



385 



60th parallel implies a change of longitude twice as great 
as the same distance along the equator. With the help of 
the slated globe, show the class how short the degrees of 
longitude necessarily are in the neighbourhood of the Pole. 

We give a suitable figure* for showing the general case. 
If R be the radius of the equator, and r the radius of the 




parallel of latitude A, passing through a given point, then 
r = R cosA. 

Q = Lat. 0, long. 0. 

P' = Lat. X, Long. 0. 

V = Lat. X, Long. P'V west. 

T = Lat 0, Long. QT west. 
Difference of longitude of M and V = arc MV = Z.MKV = LI. 

Give the boys the little problem to prove that the length 
of 1 minute of longitude measured along a parallel of lati- 
tude A is, 1 nautical mile X cosX. 

* The figure is designed to show merely the main geometrical facts. When the 
boys are familiar with these facts, the correct notation of polar co-ordinates, and the 
accepted astronomical sign convention, should be introduced. 

(E291) 26 



386 



CRAFTSMANSHIP IN MATHEMATICS 



It requires very little skill in soldering to make a wire 
model, and then the demonstration is exceedingly simple. 

In spherical geometry and trigonometry, good figures are 
essential, or very few boys will understand the problems 
considered. Here is an example of a problem from one of 
our very best books on the subject. We reproduce the original 

figure. 

The excess of the sum of the 
three angles of a spherical triangle 
over two right angles is a measure 
of its area. 

Let ABC be a spherical tri- 
angle; then, since the sum of the 
three spherical segments (lunes) 
ABA'C, A'BC'B', ACBC', ex- 
ceeds the hemisphere ACA' by 
the two triangles ABC, A'B'C'; 
and since, 

(i) the measures of the three 
spherical segments are, respec- 
tively, the angles A, B, C, of the spherical triangle, 
(ii) the measure of the hemisphere is 2 right angles, 

/. the sum of the three angles exceeds 2 right angles. . (i) 

If A is the number of degrees in the angle A, S the surface 

A 

of the hemisphere, the area of the spherical segment . S; 

180 

.'. since ABC is equal to its symmetric triangle A'B'C', the 
result of (i) is that if S is the area of the spherical triangle, 




/A + B + C 
I 180 



= 2S, 



or 



L = A + B + C -- 180 

360 ' ' 



i.e. the area S is proportional to the excess of A + B + C 
over 2 rt. Zs. 



SPHERICAL TRIGONOMETRY 



387 



I have given this theorem to boys on several occasions, 
but they have almost invariably failed to visualize the figure 
properly. They failed to pick out the spherical segments. 
We append four new figures. The first shows the spherical 
triangle plainly; the next three show the three lunes, separately 
shaded. The real trouble is that half the second lune (iii), 
viz. the part A'B'C' (= the symmetric triangle of ABC) is 
not visible. When the shaded lunes of ii, iii, iv are added 




together, it is seen that the A ABC is included twice and 
the hidden A'B'C' once. Hence the sum of the three shaded 
areas exceeds the hemisphere by the two triangles ABC 
and A'B'C'. 

Books to consult: 

1. Spherical Trigonometry, Murray. 

2. Practical Surveying and Elementary Geodesy, Adams. 



CHAPTER XXVIII 

Towards De Moivre. Imaginaries 



Interpretation of V 1 

" Please sir, what is the good of De Moivre's theorem? 
What is it really all about? What is the use of talking about 
imaginary roots to equations?" 



388 CRAFTSMANSHIP IN MATHEMATICS 

Thoughtful boys often ask such questions. It is our business 
to see that our answers satisfy them. 

The symbol V 1, if interpreted as a number, has no 
meaning. But algebraic transformations which involve the 
use of complex quantities of the form a + bi (where a and 
b are numbers, and /= V 1) yield propositions which do 
relate purely to numbers, and those propositions are now 
known to be rational and acceptable 

Boys should understand that algebra does not depend on 
arithmetic for the validity of its laws of transformation. If 
there were such a dependence, it is obvious that as soon 
as algebraic expressions are arithmetically unintelligible, all 
laws respecting them lose their validity. But the laws of 
algebra, though suggested by arithmetic, do not depend on 
it. The laws regulating the manipulation of algebraic symbols 
are identical with those of arithmetic, and it therefore follows 
that no algebraic theorem can ever contradict any result 
which could be arrived at by arithmetic, for the reasoning 
in both cases merely applies the same general laws to different 
classes of things. If an algebraic theorem is interpretable 
in arithmetic, the corresponding arithmetical theorem is 
therefore true. Sixth Form boys seem to gain confidence 
when once they realize that algebra may be conceived as 
an independent science dealing with the relations of certain 
marks conditioned by the observance of certain conventional 
laws. 

It is true that the present-day use of imaginary quantities, 
in accordance with the authoritative interpretation now 
given them, does not involve any sort of contradiction and is 
therefore presumably valid, for absence of logical contradic- 
tion is certainly a good test of valid reasoning. But Mr. 
Bertrand Russell is perhaps going a little far when he 
says (Prin. of Maths., Vol. I, p. 376) that the theory of im- 
aginaries has now lost its philosophical importance by ceasing 
to be controversial. There is still a hesitancy in the treat- 
ment of the subject in Sixth Forms, which suggests that in 



TOWARDS DE MOIVRE. IMAGINARIES 389 

the minds of at least some teachers there is a lingering doubt 
about the accepted interpretation. 

Let the early treatment of the subject be frankly dogmatic. 
Let discussions as to validity stand over for a while. 

Define the symbol V 1 merely as an expression, (1) 
the square of which = -1, and (2) which follows the ordinary 
laws of algebra. And deduce the inference that since the 
squares of all numbers, whether positive or negative, are 
always positive, it follows that V 1 cannot represent any 
numerical quantity. 

Deduce the further inference that, since V a 2 V 1 X a 2 
== V- 1 X #, V a* cannot represent any numerical quan- 
tity. Thus V I X a may be called an " imaginary " 
expression. It therefore follows that such a statement as 
A + BV 1 = a + bV 1 can only be true when A = a 
and B = b. _ 

Numbers like a + bV 1, where a and b are real numbers, 
which consist of a real number and an imaginary number 
added together, are called complex numbers. 

At this stage it is advisable to revert to the significance 
of ordinary negative quantities. If +a indicates a certain 
number of linear units in some chosen direction, a indicates 
the same number of linear units in the same line but in the 
opposite direction. Hence when working out, with algebraic 
symbols, a problem concerning distance, we interpret the 
minus symbol to mean a complete reversal of direction. 

It is desirable to take some little trouble to convince the 
pupils that, on the face of things, there is nothing in the 
expression a + b\/l to make it more " absurd " than in 
an expression like x. The result symbolized by b a 
where b is less than a is certainly " imaginary ", unless we 
add to the conception of magnitude, which necessarily 
belongs to it as a number, the further conception of direction. 

Quantities which contain V 1 as a factor are obviously 
in some ways very different from quantities which do not 
contain it. 



390 CRAFTSMANSHIP IN MATHEMATICS 

What interpretation, then, can be given to the result of 
multiplying a distance by V 1? Argand put forward an 
ingenious hypothesis, which has now received general 
acceptance. 

As we have seen, the effect of multiplying a distance by 

1 is to turn the distance through two right angles. 

Hence, whatever interpretation we give to V 1, it 
must be such that the multiplication of a distance by 

V^ X V^-l, 

i.e. by 1, must have the effect of turning a distance through 

two right angles. 

Thus it seems worth while to consider how far we may 

interpret the effect of multiplying a distance by V 1, by 

supposing that it turns the distance 
through one right angle. Evidently 
we have to devise some scheme by 
which a reversal of direction will 
be effected in two identical opera- 
tions. 

One possible plan is to revolve 
OH through a right angle either in 
the direction of S or in the direc- 
tion of T, for each of these opera- 
tions, if repeated, would bring H 

into coincidence with K. Further double applications of 

the same operation would successively bring the point to 

H(+l), to K( 1), to H again, and so on indefinitely. 

Clearly the two algebraic operations which, by definition, 

must produce, when applied in this way, the sequence +1, 

1, +1, 1, ..., represent a repeated multiplication, 
either by +V 1 or by V 1. 

Thus for exactly the same reason that we identify -1 
with a unit step taken along a line in a reverse direction to 
the unit represented by +1, we may identify +V 1 with 
the revolution of a line through a right angle in one sense, 




TOWARDS DE MOIVRE. IMAGINARIES 



391 



and V 1 with an equal revolution in the opposite sense. 

This is the accepted interpretation. 

OS is regarded as the / (or V 1) direction, and OT as 
the i (or V 1) direction. 



Complex Numbers 

Revise the early work on the significance of co-ordinates. 
Given a fixed line OA, and a fixed origin as at O, there 
are two convenient ways of fixing the position of a point P. 

1. Rectangular co-ordinates: Op 5, Pp = 2. 

2. Polar co-ordinates: Z.AOP = 22, OP 5-4. 
Evidently we may regard the rectangular co-ordinates as 

specifying not merely measurements which define the position 

of P, but also move- n 

ments by which P could 

be reached from O. The 

two movements would 

be, one of +5 along 

OA and one of +2 at 




right angles to OA. . 

The polar co-ordinates 

specify much the same thing, though in a different way. If 
to begin with we are at O and facing A, then the polar co- 
ordinates may be taken as instructions, first to turn through 
an angle of 22, and then advance along OP a distance of 5-4. 

If along a straight line a point takes two successive move- 
ments OA, AB, the length and 
direction of OB is the algebraic 
sum of the two movements. 

If OA and OB are straight 
lines or vectors which represent 
two movements not in the same 
straight line, the directed line OB 
which closes the triangle OAB 
may again be called the " sum " of OA and OB, since it 




B 



Fig. 236 



392 CRAFTSMANSHIP IN MATHEMATICS 

represents the single movement equivalent to the combin- 
ation of the two movements. Thus in fig. 225, OP may be 
called the sum (more fully, the vector sum) of the movements 
+5 along OA and +2 at right angles to OA. But if the 
movement OP be represented by the symbol R, we cannot 
in this case write R = (+5) + (+2), for this would repre- 
sent a movement of +7 from O along the line OA. But we 

p may still represent R as 

a sum, provided we do 
/ \ something to indicate 

that the component move- 
ments are at right angles. 

Ui , 5 [p A F r this purpose the 

Fig. 227 l etter * is prefixed to 

that directed number 

which represents the component at right angles to the initial 
line. This is in accordance with our interpretation of A/ 1. 
Thus the movement of OP would be represented by the 
notation (+5) + i(+2). (Fig. 227.) 

Of course, if P is confined to the line OA, a single directed 
number will suffice to define its position after a series of 
movements. But if P is forced to move about over the whole 
plane of the paper, its position may be fixed just as definitely 
by such an expression as, say, (13) + *'(+21). 

Thus we may regard an expression of the form a + ib 
as a complex number which serves to fix the position of a 
point in a plane, just as the simple number a or b fixes its 
position in a straight line. 

But bear in mind that the term " complex number " is 
only a convenient label, suggested by analogy; a + ib is 
not really one number but a combination of two numbers, 
together with a symbol i which stands for no number at 
all. The symbol i is merely a direction indicator to 
show that the movement or measurement represented by the 
second number of the complex number is at right angles 
to that represented by the first. 

Let a, b be the rectangular co-ordinates, and r, a the 



TOWARDS DE MOIVRE. IMAGINARIES 



393 



a 

Fig. 228 



polar co-ordinates of a point P. Then, since a = r cosa 
and b = r sina, 

a + ib = r cosa + i(r sina). 

Again, let P' be the point on 

OP at unit distance from O. 

Then the movement OP' may 

be represented by the complex 

number, cosa + / sina. But 

since r steps, each of length OP', would carry a point from 

O to P, we may write: 

a + ib = (cosa + /sina) X r, 

or, more conveniently, 

a + ib = r(cosa + i sina). 

It follows that we may write: 

(r cosa) -f i(r sina) = y(cosa -J- i sina). 

The conclusion is important, for it shows that we may, at 
least in this connexion, proceed just as if i stood for a number, 
Otherwise we could not legitimately assume that the twc 
expressions are equivalent. 

Note that the non-directed number r is called the modulus 
of the complex number a + ib, and the angle a its amplitude 

The operation which carries OA from its original position 
to OB, then to OC, then to OP, in equal jumps, may be 
looked upon as the repetition of a con- _ p 

slant factor, viz. a factor of the form 



cosa + V^OL sina, i.e. cosa -f / sina, 

where a is the constant angle between 
the rays from O. Since two rays 
divide the /.AOP into three equal 
parts, we may infer that m 1 rays 
would divide it into m equal parts. 




Fig. 229 



394 CRAFTSMANSHIP IN MATHEMATICS 

Hence, if ZAOP = 0,0 = ma. Since OA = r, the line OP 
may be represented by the expression: 

r(cosO -J- i sinO) 

(or, by a + $, where a = r cos# and b = r sin#). 

Again, since the factor cosa + i sina represents the turning 
of the line from its original position OA through the angle 
a, the factor 

(cosa -j- tsina) x (cos(3 + tsinp) 



must, presumably, represent a turning through the angle 
(a + j8), and therefore be equivalent to the factor 

cos (a + P) + i sin (a + P). 

Obviously, then, the identity 

coswa -j- i sin;wa (cosa + tsina) w 

is foreshadowed. The usual sequel is obvious and simple. 

Practice in the addition and subtraction of complex 
numbers is desirable; it is quite easy. Devise examples to 
enforce the notion that i is just a direction indicator, pro- 
viding us with a simple means of fixing a point P anywhere 
in a plane containing an initial line; that it serves to show 
that the second element of a complex number is at right 
angles to the first. Practice in multiplication and division 
should follow; this is also quite easy, once the boys see that 
cosa + i sina is merely a " direction coefficient ", i.e. a 
complex number which, when it multiplies another number, 
produces a result which corresponds to the turning of a line 
through the angle a. De Moivre easily follows. 

The term " imaginary number " is not a happy one; 
V 1 is just a symbol which can be treated in certain cases 
as if it were a number. In the complex number a + 16, 
a is often called the real, and ib the imaginary part. 

The fruitful suggestion was made by Gauss that instead 
of calling +1, 1, and V 1, positive, negative, and 
imaginary units, we should call them direct, inverse, and 



TOWARDS DE MOIVRE. IMAGINARIES 395 

lateral units. To Gauss the radical difference between a 
complex number and a rational number was that while the 
latter denotes the position of points along a line, the former 
denotes the position of points in a plane. 

a -f- AV 1 must be regarded as the typical number of 
algebra, " real " numbers being merely special cases in 
which b = 0. If we are confined to real values of the variables 
in y = f(x), we must admit that in the case of most functions 
there are either values of x to which no values of y correspond, 
or values of y which are not produced by any value of x. 
But if the variables are complex numbers, these exceptions 
never occur. To a value of x of the form a + b\/ 1, there 
corresponds, in the case of every possible function, a value 
of y of the form A + BV 1, #> b, A, B being themselves 
real numbers. 

The principle is so important that it must be understood 
thoroughly by all pupils. Emphasize strongly the fact that 
real numbers correspond to points in a straight line, complex 
numbers to points in a plane. If we represent the values 
of x by points in one line, and those of y by points in another, 
we cannot say that any function y f(x) establishes a one- 
to-one correspondence between all the points on the two 
lines; in most cases, whole stretches of points will remain 
outside the correspondence. But if we take two planes, 
and represent the values of x by the points of one of them, 
and the values of y by points of the other, we then obtain, 
in every function, a one-to-one correspondence between all 
the points in the two planes. This is the key to the secret 
of quadratic equations with " imaginary " roots. 



Quadratic Equations and (so-called) Imaginary 

Roots 

Complex numbers can be used to explain certain diffi- 
culties met with in the study of quadratic equations. Consider 
the example x 2 Gx + 34 = 0; the roots of which are 



396 



CRAFTSMANSHIP IN MATHEMATICS 



sometimes said to be 3 V 25. But x 2 &x + 34, 
i.e. (x 3) 2 + 5 2 , cannot be factorized; hence (we usually 
argue) there is no value of x for which y (in y = x 2 6# 
+ 34) is zero; in other words, the equation has no real 
roots. Another way of stating this is that the parabola 
y = x 2 6x + 34 has no points below y = 25 and there- 
fore does not cross the axis of x. Here is a graph of the 
function: 



(-2 




\ 






70 


















/ 




,6 


,\ 
























/ 






\ 


\ 


_> 

i40 
















/ 
















/ 










( 


),3< 


i\ 

30 


\ 








J 


29 


)- 

















X 


^^ \ u ~*> 


y 










?0 






(3,2 


^ 




















10 


- 


- 








































-5-3 O ^- +5 +IO 

i 

Fig. 230 



#- 



34 



X - 


-3 


O 


+ 3 


+ 5 


HO 


re* - 


9 


O 


9 


25 


IOO 


-GX- 


id 


O 


-18 


-30 


-60 


34 - 


34 


34 


34 


34* 


34 


Y - 


61 


34 


25 


29 


74 



X 



But if i be treated as a number whose square is 1, we may 
write, 

(x - 3) a + 5 2 = (x - 3) 2 - f* . 5 2 

= (? - 3 -f- 5/) (* - 3 - 50. 

Apparently, then, y = if jc = +3 5i. 

It is usual to say that these values are " imaginary roots " 
of the equation, or that they describe imaginary points where 
the parabola may be supposed to cross the axis of x. 

But from what we have already said about the nature of 
i, there is clearly an alternative way of regarding this, a way 
much more rational. The values +3 5i describe points 
not on the axis of x, but elsewhere in some plane containing 



TOWARDS DE MOIVRE. IMAGINARIES 397 

that line. It is obvious that it cannot be the plane of the 
paper, and we must therefore look for points in the plane 
which is at right angles to the plane of the paper. 

The necessary figure (231) consists of two parabolas, each 
y = x 2 6# + 34, head to head, with a common axis but 
in two planes at right angles to each other. A suitable sketch 
is a little difficult to make, but it may be done in this way. 
Let ABCD, EFGH be a rectangular block with square ends. 
Bisect the block by the mid-perpendicular planes JKLM, 
NPQR, STUV. The first and second intersect in the line 
ab, of which V is the mid-point. In the horizontal plane 
fliLM, draw the parabola y = x* fix + 34, with vertex 
at V. In the vertical plane o/TS, draw the same parabola, 
also with its vertex at V. The line mVn is the common axis 
of both parabolas. The heavy lines in the plane JKLM 
(xQx and Oy) are the co-ordinate axes of the primary parabola 
in the horizontal plane. The axis of the parabola intersects 
the x axis in z. As in the previous figure, Oz = 3, zV 25. 

If, instead of y = (x 3) 2 + 5 2 the parabola was 
y = (# 3) 2 , the parabola would touch the axis of x at z 
(= + 3), but when the parabola moves into the position 
y = ( x 3)2 _j_ 52^ i ts vertex is at V, (5) 2 units from the 
x axis. Hence, the points given by the complex values of 
x answering to y are at a distance 5 above and below the 
plane of the primary parabola, and on a similar parabola to 
the first, viz. the parabola in the vertical plane. Evidently 
the points are on a line through z, m' , and n' y each 5 units 
from z. 

Thus, when we take into account complex values of 
x, the complete graph corresponding to real values of the 
function y (x 3) 2 + 5 2 is not one parabola but two, 
lying in two planes. The parabola in the perpendicular plane 
contains all points answering to complex values of x which 
satisfy the given relation. Figs. 230 and 231 should be 
compared. 

Note that the line y = 25 lies in the plane NPQR, which 
is tangential to both parabolas. 



398 CRAFTSMANSHIP IN MATHEMATICS 




TOWARDS DE MOIVRE. IMAGINARIES 399 

Note also the difference between these two equations: 



* 2 - 6* - 16 = 0. 
x 2 6x + 9 = 25, 



3 + 5. 



* 2 - 6* + 34 = 0. 
/. x 2 - 6x + 9 = -25, 
/. (96 - 3) 2 = + V -25 = + 16, 
x = 3 + *5. 




(V25) 




Fig. 233 



To obtain the points on the 
curve we proceed from the origin 
to z, -f-3 units away, in the x axis, 
and then, also in the x axis, we 
proceed from z, +6 and 5 units, 
and so reach the points +8 and 
2. The vertex is 25 units below 
the x axis (see fig. above). 



The journey is a journey in 
one line, the x axis. The two 
5's are measured from z. 



To obtain the points on the 
curve, we proceed from the origin 
to s, -|-3 units away, in the x axis, 
and then we proceed +i5 and 
i5 units from ar, i.e. +5 and 5 
units in a plane perpendicular to 
the plane of the parabola, where 
we reach points on a similar para- 
bola in this new plane. 

The vertex of the primary 
parabola is 25 units above the x 
axis (see fig. above and fig. 231). 

The journey is a journey in 
two lines perpendicular to each 
other. The two 5's are measured 
from z as before, but in a perpen- 
dicular plane. 



400 CRAFTSMANSHIP IN MATHEMATICS 

Unless provided with a wire model (fig. 231), or with a 
really good perspective sketch, boys are apt to be puzzled 
by this problem. A model is much to be preferred; then 
the effect of increasing and decreasing the distance of the 
primary parabola from the x axis is easily observed. 

Warn the boys not to be led away by the remarkable 
and perfectly logical consistency of the hypothesis concerning 
V 1. It is only an hypothesis after all. Still, it is not 
advisable for learners to talk about " imaginary " roots of 
equations but rather to explain such roots in the light of 
the hypothesis in question. 

We have touched upon vector algebra. The subject 
receives considerable attention in Technical Schools but 
very little in Secondary. This is a pity, for it is a cunningly 
wrought instrument and is as useful as it is illuminating. 
Quite the best introduction to it is Part I (Kinematic) of 
Clifford's Elements of Dynamic. The first two parts of the 
book, Steps, and Rotation, should be read by all teachers of 
mathematics, and the third part, Strains, by all teachers of 
mechanics. Maxwell's Matter and Motion is a little book 
dealing admirably with the same subject. Henrici and 
Turner's Vectors and Rotors is also useful. 



TOWARDS THE CALCULUS 401 

CHAPTER XXIX 

Towards the Calculus 

Co-ordinate Geometry 

Teachers differ in opinion whether the calculus should 
be preceded by a course of co-ordinate geometry. Certainly 
anything like a complete course of co-ordinate geometry is 
not a necessary preliminary. On the other hand, some little 
knowledge of its fundamentals is advisable, and this is easily 
developed from the previous knowledge of graphs. The 
notion of the differential coefficient is nearly always made 
to emerge from considerations of the tangent to the parabola, 
but, more frequently than not, the common properties of 
the parabola have not previously been taught. This partly 
explains the haze which often enshrouds the notions under- 
lying the new subject. 

A minimum of preliminary work in co-ordinate geometry 
may be outlined. 

The boys already know that y mx + c represents a line 
making an angle whose tangent is m with the axis of x\ that, in 
short, m represents the slope of the line; and that, in whatever 
other form the equation may be written, it may be re-cast 
into the y mx form, and its slope be determined at once. 

X *V 

For instance, the intercept form ~ + ~ = 1 mav be written 

3 



y = ^/%x -f 3, and the slope is seen to be \/3. 

The boys must be able to determine the equation of a 
line satisfying necessary conditions. They already know that 
if they are told a straight line must pass through a given 
point, this condition alone is not enough to determine the 
line, since any number of lines may pass through the point. 
But if they are given some second condition, e.g. the direction 
of the line, or the position of a second point through which 

(291 27 



402 



CRAFTSMANSHIP IN MATHEMATICS 



it passes, then the two data completely fix the line. This 
fits in with the fact that the equation of the line must contain 
two constants. Make the boys thoroughly familiar with the 
ordinary rules for finding the equation of a line satisfying 
two given conditions. 

Types of suitable exercises for blackboard oral work: 

1. Find the equation of a straight line cutting off an 
intercept of 2 units' length on the axis of y and passing through 
the point (3, 5). 

2. Find the equation of a straight line drawn through 
the point (3, 5), making an angle of 60 with the axis of x. 

3. Find the equation of a straight line passing through 
the points (2, 3), (-4, 1). 

The last exercise is a type with which the boys should be 
thoroughly familiar. It will be required often in future 
work. The equation should therefore be familiar in its 
general form, and should be illustrated geometrically. 

Find the equation of a line passing through the two points 
A(#i, jVi) and B(# 2 , y 2 ). We may proceed in this way: 



W-T 




Suppose y = mx 
m and c are unknown. 



c represents the equation, where 



The particular point (# t , y x ) is on the line, .'. y l mx l + c. (i) 
The particular point (x 2 , y 2 ) is on the line, .'. y% = mx 2 -f- c. (ii) 
The point (#, y), any point, is on the line, .'. y = mx + c. (iii) 



TOWARDS THE CALCULUS 403 

Subtracting (i) from (iii), 

y y 1 = m(x xj. 

Subtracting (i) from (ii), 

^2 y\ = "K*2 *i)- 
.'. by division 



or (y - yi ) - 



which is the required equation, -^ - ~ representing m 
(the slope) in y = mjc. ^ 2 ~ ^ 

Beginners rarely see this clearly, unless the algebra is 
clearly illustrated geometrically. From the last figure, 




take out the two similar triangles, and show the lengths 
of their perpendiculars and bases in terms of co-ordinates. 
The slope of the line AB is given from the first triangle; 

= 9* i-J, A and B being the two specified points on 

AK # 2 ~" #1 
the line. 

The slope of the line AB is also given from the second 

triangle, - = y ~~ ?\ A being a specified point, and P 
AM x #! 

being any point on the line. 



404 CRAFTSMANSHIP IN MATHEMATICS 

But the triangles are similar, and the slopes are therefore 
identical. 

.- y ~~ yi = yLny\ 

X X l X 2 Xj* 

or (y y x ) = ^-"H^ 1 (x x^, as before. 

# 2 Xi 

Or again: take the slope found in the last case, say , 
and substitute for m in y ~ mx + c. We have * 2 ~~ x * 



a) 



and since the line passes through (x ly j/ x ), [or (# 2 , y 2 ) might 
be chosen if preferred], 



By subtraction, we have from (i) and (ii) 

~ x \)t as before. 



Every step must be substantiated geometrically. It is 
fatal to allow the boys to look upon the problem as mere 
algebra. The boys should be able to write down instantly the 
equation of a line passing through two points and under- 
stand its full significance. But in evaluations of this kind, 
do not be satisfied with just the typical textbook formal 
solutions. Vary the work. 



The Parabola and its Properties 

Throughout the teaching of co-ordinate geometry, let all 
principles be established first by pure geometry. Let the 
picture come first, and teach its new lessons. Then let 
symbolism follow. Geometry treated as pure algebra tends 
to lose all semblance of its essential space relations. At 



TOWARDS THE CALCULUS 



105 



least the parabola, if not the ellipse, will already have received 
some attention. It will have been touched upon in connexion 
with graphs and quadratic equations, and the boys may have 
learnt something about the paths of falling bodies; they 
may also know that in certain circumstances the chains of a 
suspension bridge, and vertical sections of the surface of a 
rotating liquid, form parabolas. 

It is useful to give the boys a mechanical means of readily 
drawing a parabola. It saves much time, and encourages 
them to use good figures. Here is one way. 

An ordinary T-square slides along AB, the left-hand edge 
of a drawing board, in the usual way, the edge AB answering 




C 



Fig. 235 



as a directrix. A string equal in length to KG is fastened 
at G, and at a fixed point S in a line XX perpendicular to 
AB. A pencil point P keeps the string stretched and remains 
in contact with the edge KG of the T-square. As the T-square 
slides up and down the edge of the board, the pencil traces 
out a parabola with focus S. 

A parabola is the locus of a point whose distance from a 



406 



CRAFTSMANSHIP IN MATHEMATICS 



fixed point is equal to its distance from a fixed straight 
line. Help the boys to see that from this definition certain 
properties follow at once: 

(1) PM (diam.) = PS. 

(2) ES (semi-latus rectum) = EM' = SX. 

(3) AS = AX. 

(4) SX = 2SA. 

(5) ES - 2SA. 

(6) EF (latus rectum) = 4SA. 



The main property to be mastered is the slope of the 
tangent, and to this end the following summary of pre- 
liminary work is suggested. All principles should be established 

first geometrically, then 

- analytically, and the 

boys must be made to 
see that the results are 
identical. 

1 . The principal 
ordinate of any point P 
on a parabola is a mean 
proportional to its ab- 
scissa and the latus rec- 
tum. 

i.e. PN 2 - 4AS.AN. 

Analytically: call the 
point P, (x 9 y)\ SA = 
Fig. 236 AX = *(8ay). Theny 2 

- 4ax. (Fig. 236.) 

If the directrix MX is the y axis, the equation becomes 
y 2 = &a(x a). 

2. If a chord PQ intersects the directrix in R, SR bisects 
the external angle QSP' of the triangle PSQ. 




TOWARDS THE CALCULUS 



407 



Drop JLs PM 
and QV on direo 
trix. As PMR and 
QVR are similar. 



PR 
QR 



PM 
QV 



PS 
QS' 




Fig. 237 



3. If the tangent 
at P meets the direc- 
trix in R, the angle 
PSR is a right angle. 
Deduce this from 
the preceding pro- 
position. When Q 
coalesces with P, 
each of the marked 

equal angles becomes a right angle. (See next figure.) 

4. The tangents at the extremities of a focal chord intersect 
D 



PP jbcal chord 
RP, RP - tangents 
Z.PRP*- 




Fig. 238 



at right angles on the directrix, i.e. the tangents at P and P', 
the extremities of the focal chord PSP ; , make a right angle 



408 



CRAFTSMANSHIP IN MATHEMATICS 



at R, where they meet on the directrix. Observe that RS 
meets the focal chord at right angles (cf. No. 3). 

5. The subtangent is equal to twice the abscissa, i.e. TA= AN 

or, TN = 2AN. 




Fig. 239 



6. The foot of the focal perpendicular of any tangent lies 
on the tangent at the vertex, i.e. Y, the foot of the -L SY to 
the tangent PT, lies on the tangent at A. 




Fig. 340 



TOWARDS THE CALCULUS 409 

rr>i 7 r . .2 latus rectum ~ 
7. 1 he slope of any tangent = - - - - -. ror m 
^ J J 6 ordinate 

the last figure the triangles YAS and TNP are similar. Hence 
slope of tangent PT 

^ PN = SA = 2AS ^ \ latus rectum 

"" NT ~ AY ~~ PN ordinate ' 

i.e. the slope of the tangent to the axis of the parabola. If 
the figure is turned round, so that the slope of PT is to the 

ATT i 11 ordinate 

tangent AY at the vertex, then slope = - -- . 

$ latus rectum 

How may this slope be expressed in rectangular co- 
ordinates? The equation of a secant cutting the curve in 

p *> and Q is 



If a figure be drawn accurately (this is not easy), actual 
measurement will show that the slope of the secant is the 
ratio 

\ latus rectum^ 
mean of ordinates 

To obtain the equation of the tangent at (x lt y^ we take 
Q indefinitely close to P, so that ultimately j> 2 = y v The 
equation to the secant then becomes: 



a, , 
or y = (x + *! 



which is the equation to the tangent, and the slope of the 

* *u 20 . 4 latus rectum u - 
tangent is thus , i.e. = - - - , as before. 
y l ordinate 



410 CRAFTSMANSHIP IN MATHEMATICS 

The Tangent to the Parabola 

If future work is to be understood, the tangent to the 
parabola, and its various implications, must receive close 
attention. The necessary further elucidation may thus be 
summarized. 

1. To find the condition that the straight line y = mx -f- c 
may touch the parabola y 2 * kax. 

Since y = mx -f- c, 

.'. y 2 - (mx + c)*; 
and since y 2 lax, 
:. (mx + f) 2 = lax. 

By solving this equation we obtain the abscissae of the two 
points in which the straight line cuts the curve. The line 
will touch the curve if the two points coincide, and the con- 
dition for this is that the roots must be equal, 

i.e. in m 2 * 2 + 2x(mc - 2a) + c 2 = 0, 

l(mc - 2a) 2 = Im 2 c 2 9 

a 

i.e. a = me, or c . 
m 

Hence the line y = mx + c touches the curve y 2 kax if 

c = _ (where m is the slope which the tangent makes with 
m 

the axis). 

2. To find the point where the tangent y = mx + touches 
the parabola y 4a#. 

As before, 

(mx + c) 2 = lax, 

/. [mx -j- ) == 40#, 
> m' 



( 



or \ mx 1 = 



and since y 2 = lax, y = . 



TOWARDS THE CALCULUS 411 

a 2a\ 



Hence the point required is ( -, ). 

\m 2 m/ 



3. Compare the two forms of the equation of the tangent, 



viz. yy = 2a(x -f- #i)> and y = mx + . 

m 



The first may be written, 



2a 

y = x 



Hence we may write the two forms in parallel thus: 



y = mx + - 
nt 



TU * *u r . a , 

They represent the same line; . . m, and 

Ji i 

That these two last equations are consistent may easily be 
shown by evolving the second from the first, the connecting 
link being y^ = 



<0 t_ 

Evidently, then, the equation y = mx + 1S the tangent 

o ^^ 

at the point (x l9 y^, i.e. ( , ). 

\m m / 

4. Verify geometrically that the tangent y = w# + 

9 Wl 

touches the parabola y 2 = &ax at the point ( , j. (This 
verification is of great importance.) m 

(The abler boys ought to do this without any further 
help). 

AY = tangt. at vertex = y axis. 
AX = x axis. 
S = focus. 

TA = AN (subtangent = 2 abscissa) 

SZ meets the tangent PT at rt. Z.s at Z, since AZ is the tangent at 
vertex. 



412 



CRAFTSMANSHIP IN MATHEMATICS 



The rt. angled As TAZ, ZAS, TZS, TNP are all similar. 
Since TA = TN, /. ZA = PN. 
AS = a = dist. of focus to vertex. 




Fig. 241 



AZ = = intercept on y axis. 
m 

PN 

- = m slope of tangent. 

TA _ AZ _. _ AZ 2 
_= _,or I*-, 



/. AN 



AZ 2 
AS 



a 



= abscissa of P, 



and 



2a 



PN = 2ZA = = ordinate of P. 



Q.E.D. 



Observe, again, that the slope of the tangent 
__ \ latus rectum __ 2a 



ordinate 



- = m. 
2a 



m 



This pictorial parallelism between the geometry and 
algebra is essential whenever it is possible. Let the boys 
see that co-ordinate geometry is geometry and not mere 
algebra. But of course the geometrical figure is also a 
graph, to be interpreted algebraically. 



We have taken the subject of co-ordinate geometry 



TOWARDS THE CALCULUS 413 

thus far, less for its own sake than as an introduction to the 
next chapter. Co-ordinate geometry is an easy subject to 
teach, and boys like it, provided the geometry itself is made 
clear. As a subject of mere algebraic manipulation, un- 
associated with pure geometry, its value is slight, and time 
should not be spent over it. 

Methods of Approximation 

The calculus is such a valuable mathematical weapon, 
and the fundamental ideas underlying it are so simple, that 
the subject should find a place in every Secondary school. 
It might be begun in the Fifth Form, if not in the Fourth, 
though naturally the first presentation must be of a simple 
character. This simple presentation is easily possible. The 
more technical side of the subject, as elaborated in the 
standard textbooks, is wholly unnecessary in schools. 

It was, I believe, Professor Nunn who pointed out that 
the history of the subject suggests the best route for teachers 
to follow. Although Newton and Leibniz are rightly given 
the credit of being the creators of the calculus as a finished 
weapon, the preliminary work of certain of their predecessors, 
especially Wallis, from which the main idea of the calculus 
was derived, must always be borne in mind. Wallis's work 
is merely a special kind of algebra and may readily be under- 
stood by a well-taught Fourth Form. 

If we are thus to begin with approximation work, there 
is much to be said, as pointed out by Professor Nunn, for 
beginning with integration rather than with differentiation. 
The necessary arguments are so simple and the results 
so valuable that the rather radical departure from normal 
sequence is justified. For all practical purposes, Wallis 
was the actual inventor of the integral calculus, and Wallis 's 
own work and methods serve to give young pupils a clear 
insight into the new ideas. 

This early work, in differentiation as well as in integra- 
tion, should be taught as a calculus of approximations. The 



414 CRAFTSMANSHIP IN MATHEMATICS 

pupils should learn that such investigations give results 
which may be regarded as true to any degree of approxima- 
tion, though not absolutely true. When later the calculus 
itself is formally taken up, and the pupils are able to grasp 
the modern theory of " limits ", they should be able to see 
that the new arguments, if properly stated, do as a matter 
of fact give results which are unequivocally exact. They 
must not be allowed to assume, at that later stage, that the 
arguments of the calculus prove merely approximately true 
results, and yet that these may be treated as if they were 
exact truths. This illegitimate jump from possible truth 
to certain truth is often made, it is true; but the deduction 
commonly involves the fallacious use of such terms as 
" infinitely small ", " infinitely great ", and the like. 

" Methods of approximation are inferior methods and 
do not yield exact results." Granted. But these methods 
are best for beginners, if only because they form a good 
introduction to the exact methods of the calculus, and they 
are, after all, based upon a kind of reasoning which is rigor- 
ous enough for practical purposes. But the important thing 
is to make the pupils feel that they must never be finally 
satisfied until they have mastered a method which yields 
results that admit of no doubt at all. 

The beginner has already learnt, or should have learnt, 
from his graph work the main idea of the real business to 
be taken in hand, and that is the nature of a function: that 
the value of one variable can be calculated from the value 
of another by the uniform application of a definite rule 
expressed algebraically. 

We append a few suggestions for work in suitable approxi- 
mations. 

1. Rough approximations. 

(i) Revise certain exercises in mensuration, e.g. find the 
area of a circle and of some irregular figures by the squared 
paper method. 

(ii) Surveyor's Field-Book exercises; measure up some 



TOWARDS THE CALCULUS 415 

irregular field, or other area, but insist that all such results 
are only rough approximations. 

2. Closer approximations, and the methods involved. 

(i) Revise the work on expansion (in physics). For instance, 
the coefficient of linear expansion of iron is '00001. Justify 
the rule of accepting -00002 instead of (-00001) 2 for area 
expansion, and -00003 instead of (-00001) 3 for cubical expan- 
sion. Show the utter insignificance of the rejected decimal 
places. Refer to the geometrical illustrations of (a + b) z 
and (a + ft) 3 . 

(ii) Estimate the area of a triangle as the sum of a number 
of parallelograms. The more numerous the parallels and the 





z w . 

Fig. 243 

more numerous the parallelograms, the more negligible do 
the projecting little triangles become. Observe that if the 
number of parallelograms is doubled, each shaded triangle 
is reduced to one-fourth; and so on. 

(iii) Estimate the volume of a pyramid as the sum of the 
volumes of a number of flat prisms, gradually diminished 
in thickness. 

(iv) Estimate the volume of a sphere regarded as the sum 
of a number of pyramids formed by joining the centre of the 
sphere to the angular points of a polyhedron, the number of 
whose faces is increased indefinitely. The pyramids formed 
have as their bases the faces of the polyhedron; the volume 
of each pyramid (face X height) /3, hence the volume of 
the sum of the pyramids = (sum of faces) X height/3. If 
the number of faces be increased, the sum of the faces becomes 
more nearly equal to the area of the spherical surface, and 
then the height of the pyramid is more nearly equal to the 



416 CRAFTSMANSHIP IN MATHEMATICS 

radius r of the sphere. But the sum of the faces can never 
be quite equal to the surface of the sphere, though we can 
so increase the number of faces of the polyhedron that the 
approximation may be closer than any degree we like to 
name. The spherical surface is necessarily greater than the 
sum of the flat faces of the polyhedron and can never be 
reached: it is an unreachable limit. If the sum of the faces 
could become equal to the surface of the sphere the sum 
would be 4:irr 2 and then the height of each pyramid would 
be equal to r. Hence the volume of the sphere would be 

r 4 

4?rr 2 X - = -77T 3 . Now this result agrees with the result 
3 o 

arrived at by other methods, and it is correct. Still, to obtain 
the result, we had to jump from flat-faced pyramids (though 
these may have been made inconceivably small) to corre- 
sponding bits of spherical surface which were not flat. 
We have still to discover whether such a method is allow- 
able. 

(v) The value of TT. The pupils may be allowed to assume 
from a figure they will readily guess that if 2 regular 
polygons with the same number of sides be, respectively, 
inscribed within and circumscribed without a circle, the 
length of the circumference of the circle will be less than 
the perimeter of the circumscribed polygon but greater than 
that of the inscribed polygon. Show the pupils that the 
determination of TT is thus merely a question of arithmetic, 
though of very laborious arithmetic, inasmuch as we have to 
determine the perimeters of polygons of a very large number 
of sides; the greater the number of sides, the greater the 
degree of approximation of the value of TT. Give a short 
history of the evaluation of TT, from the time of Archimedes 
onwards. Point out that the irrationality of TT has now been 
definitely demonstrated, so that it is useless for computers 
to waste any more time over it. Make the pupils see that 
the method of evaluating TT is only a method of approximation, 
and that in this case no better method is ever to be hoped 
for; that we can obtain values more and more approximating 



TOWARDS THE CALCULUS 



417 



to the ratio of the circumference to the diameter, but there 
can be no final value, as the decimal can never terminate. 



Area under a Parabola 

We cut off a parabola by a line P'P perpendicular to its 
axis OA, and enclose it in the rectangle P'M'MP. We will 
calculate the shaded area OPM, i.e. the area " outside " or 
" under " the half parabola AOP. Let the parabola bey=kx 2 




We may divide OM into any number of equal parts, and 
on these parts construct a number of rectangles of equal 
breadth, set side by side as shown in the figure. The added 
areas of the rectangles are evidently in excess of the area 
OPA, but by increasing the number of rectangles indefinitely, 
the excess is indefinitely diminished. 

We will begin by dividing OM into a small number of 
parts, and then increase the number gradually. As the first 
division OQ is gradually to be diminished, we will consider 
the rectangle on it to be of zero area. Hence OR = (2 1) 
= 1 unit; .'. RR' = I 2 units, OS = (3 1) = 2 units; 

(K291) 28 



418 CRAFTSMANSHIP IN MATHEMATICS 



A PA 










L 




~\ I 
TOTAL- ARC* AM . 
DIV DEO INTO 4-3 2 

-as. 





TOTAL AREA AM 

DIVIDED INTO 3*2* 

-12 


2* 


- -4 ' 




. . _, _ s 


.av. 




R 










z 






I 2 










L |Z 







___ TOT 



M 

2 O*i25 0123* 



P A 



TOTAL AREA A M 






TOTAL AREA A M 






TOTAL AREA A M 




DIVIDED INTO 6*5 






DIVIDED INTO 7 "6 






DIVIDED INTO 8 X 7 




- 150 








252 


1 






-39E 










j 






























2 



























6 










7 


1 


2 










2 

5 












G' 














2 












2 






I 3 








FT 


4 











, 


5 






II 








i.t 










1 * 


4 








i 1 2 1 








[?[ 










P] 5 










D9RSTUM 09RSTUVM O9RST 


U V W M 
4567 



(IV) 



(v) 
Fig. 244 



.'. SS' - 2 2 units; OT = (4 - 1) = 3 units, /. TT' = 
units; &c. 

We may tabulate the results thus: 



Linear 
Units in 


Square Units of Area in 


Ratio of 
(*) to (a). 


OM. 


PM. 


(a) Rect. 
AM. 


(6) Sum of contained Rectangles. 


Lowest 
Terms. 


Re- 
written. 


3 


2 2 


3 x 2 2 


1 2 + 2 2 


A 


A 


4 


3 2 


4 X 3 2 


I 2 + 2 2 + 3 2 


A 


A 


5 


42 


5 X 4 2 


I 2 -}- 2 2 -f 3 2 + 4 2 


a 

8 


9 
24 


'6 


5 a 


6 X 6 2 


p + 2 2 + 32 + 42 + 52 


M 


a* 


7 


6 2 


7 X 6 2 


I 2 + 2 2 + 3 2 -f 4 2 -f 5 2 + 6 2 


M 


11 


8 


7 2 


8 X 7 2 


I 2 + 2 2 + 3 2 -f 4 a + 5 2 + 6 2 + 7 2 


A 


M 



TOWARDS THE CALCULUS 419 

The rewritten ratios show the numerators in A .P., and the 
denominators as multiples of 6. Obviously if m be the 

number of units in QM, the ratio may be written -- or 
1 Gm 



Thus the area of the added rectangles is equal to the 
area of the rectangle AM + a fraction depending on the 
value of m. It is easy to prove that the law holds good for 
all values of m. 

The ratio i _ enables us to write down as many 
6m J 

terms as we please. For instance if m 1000, the ratio 
= elH) o or 4 + 6 oW Hence if we built up a figure with 
1000 rectangles, the total area of the rectangles would be equal 
to of AM + a small area equal to Q-^Q of AM. 



Evidently by taking m large enough, the fraction 

6m 

becomes so small as to be insignificant, and thus the com- 
bined area of the rectangles can be made to differ as little 
as we please from | the area of AM. And as the rectangles 
are made narrower and narrower, the area they cover will 
eventually become indistinguishable from the area under 
the curve OP (fig. 243); e.g. if m = 1000 the sum of the top 
left-hand corners of the rectangles projecting outside the curve 
is only -Q-QOQ ^ AM. Finally the tops of the rectangles will 
be indistinguishable from the curve itself. We conclude, 
therefore, that this area under the curve is, at least to a very 
great degree of accuracy, | of the rectangle AM. Since 
OM = x, and PM kx* (fig, 243), we express the conclusion 
by the formula 

A = 



It follows that the area AOP is f the area AM. Hence the 
whole area of the parabola up to PT is f OA X P'P. 

A point for emphasis: " Having proved that the area 
under the curve is, apparently to an unlimited degree of 
closeness, \ of the rectangle AM, we are almost forced to 



420 



CRAFTSMANSHIP IN MATHEMATICS 



believe that the former is exactly \ of the latter." Still, the 
fact remains that what we have proved is only an approxima- 
tion. Do not disguise the theoretical imperfection of the 
conclusion. Do not slur over the fact that we have merely 
an approximation formula, though it is quite proper to em- 
phasize the other fact that no limit can be set to the close- 
ness of the approximation which it represents. 

The particular approximation result arrived at is easily 
extended. Let an ordinate start from the origin and move 
to the right. If it has a constant height, y = k, it will, in 
moving through a distance x, trace out an area, A = kx. 
If its height is at first zero, but increases in accordance with 
one of the laws y = kx, y = kx*, y ---- kx 3 , the area traced 
out will be given by the corresponding law, A \kx*> 
A = J&# 3 , A =- \kofi. These results we might establish by 
proceeding exactly as before. Calling the function which 
gives the height of the ordinate, the ordinate function, and 
the function which gives the area traced as the area function^ 
the results may be summarized simply 



Ordinate functions 
Corresponding area functions 


kx 


kx 1 
1 


kx 2 

s* 


kx 3 
4 iX 


kx n ~ l 

~kx n 
n 



This summary exhibits Wallis's Law. 

Books to consult: 

1. Teaching of Algebra, Nunn. 

2. Cartesian Plane Geometry, Scott. 

3. Modern Geometry, Durell. 



THE CALCULUS: SOME FUNDAMENTALS 421 
CHAPTER XXX 

The Calculus : some Fundamentals 

First Notions of Limits 

To boys the two terms " infinity " and " limits " are 
always bothersome, and it is doubtful if the first ought to 
be used in Forms below the Sixth. A misapprehension as 
to the significance of both terms is responsible for much 
faulty work, much fallacious reasoning. 

What is a point? " A point is that which marks position 
but has no magnitude." But how can a thing with no magni- 
tude indicate position? If it has magnitude, is it of atomic 
dimensions, say 10~ 24 cm.? Or is it 10~ 10 of this? Obviously 
if it has magnitude at all, a certain definite number side by 
side would make a centimetre. But this is entirely contrary 
to the mathematician's idea of a point. 

If a line is composed of points, the number of points 
certainly cannot be finite; otherwise, if the number happened 
to be odd, the line could not be bisected. Again, if the side 
and the diagonal of a square each contained a finite number of 
points, they would bear a definite numerical ratio to each 
other, and this we know they do not. The existence of in- 
commensurables proves, in fact, that every finite line must, 
if it consists of points, contain an infinite number. In other 
words, if we were to take away the points one by one, 
we should never take away all the points, however long we 
continued the process. The number of points therefore 
cannot be counted. This is the most characteristic property 
of the infinite collection that it cannot be counted. 

Consider two concentric circles. From any number of 
points on the circumference of the outer circle, draw radii 
to the common centre. Each radius cuts the circumference 
of the inner circle, so that there is a one-to-one correspondence 
between all the points on the outer circle and all the points 



422 CRAFTSMANSHIP IN MATHEMATICS 

on the inner. Imagine the outer circle to be so large as to 
extend to the stars, and the inner one to be so small as to 
be only just visible to the naked eye. Further, imagine 
an indefinitely large number of points packed closely round 
the circumference of the big circle, and all the radii drawn; 
the number of corresponding points on the inner circum- 
ference must be the same as the number on the outer. 
Clearly, in any line however short, there are more points 
than any assignable number. However large a number of 
points we imagine in a line, no one of them can be said 
to have a definite successor, for between any two points, 
however close, there must always be others. 

Again, consider the class of positive integers. They may 
be put into one-to-one correspondence with the class of 
all even positive integers, by writing the classes as follows: 

1|2|3|4|5|6| 

2 I 4 I 6 I 8 I 10 I 12 I 

To any integer a of the first class there corresponds an integer 
20 of the second. Hence the number of all finite numbers 
is not greater than the number of all even finite numbers. 
Evidently we have a case of the whole being not greater 
than its part. 

Thus we have another characteristic of classes called in- 
finite: a class is said to be infinite if it contains apart which can 
be put into a one-to-one correspondence with the whole of itself. 

It is possible to imagine any number of sequences whose 
numbers have a one-to-one correspondence with all the 
integers, for instance all the multiples of 3, or of 7, or of 97. 
The characteristics of all such sequences are: (1) there is 
a definite first number; (2) there is no last number; (3) every 
number has a definite successor. Hence they must all be 
supposed to have the same infinite number of members. 

It is important to notice that, given any infinite collection 
of things, any finite number of things can be added or taken 
away without increasing or decreasing the number in the 
collection. 



THE CALCULUS: SOME FUNDAMENTALS 423 

It will be agreed that the nature of an infinite number is 
beyond the conception of an ordinary boy, and the boy 
should not be allowed to use the term. Even the ordinary 
teacher may ponder over the paradox of Tristram Shandy. 
A man undertakes to write a history of the world, and it 
takes him a year to write up the events of a day. Obviously 
if he lives but a finite number of years, the older he gets 
the further away he will be from finishing his task. If, how- 
ever, he lives for ever, no part of the history will remain 
unwritten. For the series of days and years has no last term; 
the events of the nth day are written in the nth year. Since 
any assigned day is the nth, any assigned day may be written 
about, and therefore no part of the history will remain un- 
written. 

Neither Tristram Shandy nor Zeno is meat for babes, but 
there are certain elementary considerations of number se- 
quences with which boys should be familiar. 

" Number " in the more general sense means simply 
the ordinary integers and fractions of arithmetic. All numbers 
in mathematics are based on the primitive series of integers. 
A fraction is, strictly speaking, a pair of integers, associated 
in accordance with a definite law. This law enables us to 
substitute for each single integer a pair of integers which 

are to be taken as equivalent to it. Thus - is equivalent to 5. 

In this way we get an infinite number of numerical rationals 
of the same form. 

Between any two numbers of the sequence of integers, 
there is an infinite number of rationals. For instance, between 
8 and 13 there is an infinite series of rationals, or between 8 and 
9. Obviously, then, the rationals between, say, 8 and 9 form 
a sequence that is endless both ways. Between 8 and 9 we 
have, for instance, 8-5; between 8 and 8-5 we have 8-25; between 
8-25 and 8-5 we have 8-375; and so on indefinitely. Con- 
secutive fractions, that is, fractions between which, for example, 
a mean cannot be inserted, are inconceivable, just as are 
consecutive points in space. The integers 8 and 9 are the 



424 CRAFTSMANSHIP IN MATHEMATICS 

first numbers met with beyond the sequence. We call these 
numbers 8 and 9 the upper and lower limits of the sequence. 
There is no last rational less than 9 and no first rational 
greater than 8. It is erroneous to say that the terms of a 
sequence ultimately coincide with the limit. The limit is 
always outside the sequence of which it is the limit. 

Consider the sequence 2 y, 2 J, 2 J, . . . 2 -, 

n 

as n increases endlessly. Here successively higher integral 
values form a sequence of rationals which constantly rise in 
value but have no last term. There is, however, a rational 
number, 2, which comes next after all possible terms of the 
sequence. That is to say, if any rational number be named 

less than 2, there will always be some term less than 2 - 

n 

between it and the number 2. This is what is meant by 
calling 2 the limit of the sequence. 

A and B are a given distance apart, say 2". We attempt 
to reach B from A by taking a series of steps, the first step 
being half the whole distance; the second, half the remaining 
distance; the third, half the still remaining distance; and 
so on. When would B be reached? Obviously the answer is 
never. For any step taken is only half the distance still 
remaining. Thus the successive distances, in inches, are, 
1, , J, J, y, and so on. However many of these distances 
are added together, the sum would never be equal to the 
whole distance, 2". It is thus absurd to talk about summing 
a series to infinity. The limit, 2, is not a member of the 
series; it is unreachable and stands, a challenger, right out- 
side the series , as a limit always does. 

The example is well worth pursuing. 

The sum of the series 1 + 1 + I + . . . + A = 2 - Jj. 



The sum of the series l + + +...+- = 2-. 



THE CALCULUS: SOME FUNDAMENTALS 425 

We can, of course, take a number of terms of the series that 
will be great enough to make the sum fall short of 2 by less 
than any fraction that can be named, say less than 1/100,000. 

We have to calculate the value of n so that may be equal 

to or less than 1/100,000. By trial we find = and 

11 2 16 65536 



2" = 131072* The lattef is leSS than 1 / 100 ' 000 - Thus if 
we take 17 terms of the series, the sum differs from 2 by 
less than 1/100,000. 

It is impossible to take enough terms to make the sum 
equal to 2. There is always a gap l/2 n . However great n may 
be, l/2 n is always something; it is never zero. We may say 
that the sum of n terms becomes nearer and nearer 2 as n 
becomes greater and greater, or that it tends towards 2 as n 
becomes indefinitely great. Do not use the term infinity. 

Consider another example, the decimal 11111.... 

11111... = ye + Too i" Tooo" ~r Too (To ~i~ ^ c ' 

The sum of the first 2 terms = = i -. 

100 9-^- 

The sum of the first 3 terms = 



The sum of the first 4 terms 



1000 9 T | T * 
1111 _ 1 
10000 ~ 9-' 



By increasing the number of terms, the sum can be made to 
differ by less and less from |, and this difference can be made 
smaller than any quantity that can be named. Hence ^ is 
the limit to which the sum tends, though this limit can never 
be actually reached. 

So in cases like the area under a curve. Where we say that 
PM is the limit of the ordinate pm, we mean that by taking 
mM constantly smaller, pm may be brought constantly 
nearer PM, and that it never occupies a position so near that 




426 CRAFTSMANSHIP IN MATHEMATICS 

it could not be still nearer. It always remains the opposite 

side of the rectangle, and never actually coincides with PM, 

for PM stands outside all pos- 
sible positions of pm. But PM 
is the first ordinate that stands 
outside the series, that is, there 
is no other ordinate between 
PM and the series of all possible 
positions of pm. 
Fig 24S The teacher should devise 

other illustrations of the nature 

of a limit. The notion is fundamental, and the pupils must 

understand it. 

Rate 

The old geometers were concerned more with drawing 
tangents to curves, and with finding the areas enclosed by 
curves, than with rate of change in natural phenomena; 
but the latter idea as well as the former one was certainly 
in Newton's mind, and was embodied in the language of the 
calculus, as we now call it, which he and Leibniz invented. 
The two ideas, tangency and rate, are virtually just two facets 
of the same idea, and in teaching the calculus the two should 
be kept side by side. 

Pupils will have learnt something already about tangency. 
And if they have begun dynamics, as they ought to have done, 
they will have some idea of the nature of " Rate ". Rate is 
one of those rather subtle terms which are much better con- 
sistently used than formally defined. Even in the lower Forms, 
boys should be given little sums in which the term is correctly 
used: " at what rate was the car running?" and so forth. 
But before the calculus is begun, the notion of rate must be 
clarified. This means presenting the notion, in some way, 
in the concrete. Practical work is essential, even if the experi- 
ments are only of a rough and ready character. Suitable 
experiments are described in any modern book on dynamics. 

Consider a train in motion. How can we determine its 



THE CALCULUS: SOME FUNDAMENTALS 427 



velocity at some instant, say at noon? We might take an 
interval of 5 minutes which includes noon, and measure how 
far the train has gone in that period. Suppose we find the 
distance to be 5 miles; we may then conclude that the train was 
running at 60 miles an hour. But 5 miles is a long distance, 
and we cannot be sure that exactly at noon the train was 
running at that speed. At noon it may have been running 
70 miles an hour, or perhaps 50 miles, going downhill or 
uphill at that time. It will be safer to work with a smaller 
interval, say 1 minute, which includes noon (perhaps 
half a minute before to half a minute after Big Ben begins 
to strike), and to measure the distance traversed during 
that period. But even greater accuracy may be required: 
one minute is a rather long time. In practice, however, the 
inevitable inaccuracy of our measurements makes it useless 
to take too small a period, though in theory the smaller 
the period the better, and we are tempted to say that for ideal 
accuracy an " infinitely small " period is required. The older 
mathematicians, Leibniz in particular, yielded to this tempta- 
tion, and so gave wrong explanations of the working of the 
new mathematical instrument (the calculus) which they 
invented. 

Revise rapidly some of the easier 
graph work and show how change of 
rate is indicated by change of steep- 
ness in the slope. 

The careful study of a falling body 
will go far to make clear the notion of 
rate. Refer to Galileo's experiments 
on falling bodies. Generally speaking, 
it will not be possible to repeat such 
experiments, and so obtain first-hand 
data; the necessary data must therefore 
be provided otherwise. Let fig. 246# represent the path of 
a body falling from a tower or down a well. The three 
lines allow the three sets of values (distances, velocities, times) 
to be shown in parallel, the distances and velocities being 



DISTANCES VELOCITIES TIMES 
(FT) (Frpersc.) fseca) 



.64 64 



256 1(28 J 

Fig. 2460 



428 



CRAFTSMANSHIP IN MATHEMATICS 



DISTANCES VELOCITIES TIME 
(Fr) (FT parse^ (sec,) 



shown at the end of 1, 2, 3, and 4 seconds, respectively. Use 
the data to verify (perhaps in some degree to establish) the 
formulae v = ft , v = M + /*> * = i(# + *0*> 5 = /* 2 

Boys are often puzzled about the 32 (the acceleration 
constant). In the first place, it is a power of 2, and they confuse 
it with t 2 . (It is really best to use the nearer value 32-2, 
even though the arithmetic is a little more difficult.) In the 
second place the boys are apt to forget that this acceleration 
number is merely the value attached to a particular interval 
of time, viz. 1 second. They should be given a little practice 
with smaller intervals, say J seconds. 
The second figure, a modification of the 
previous figure, is therefore useful. It 
represents the happenings in the first 
second, at quarter-second intervals. 
Since the same amount of extra velocity 
is added on per second, we have to take 
one-quarter of this for each quarter of 
a second. Observe that although this 
figure really represents the happenings 
in the first second of the previous figure, 
the two figures have an identical appear- 
ance so far as the line-divisions are con- 
cerned. The one second is divided up exactly as the four 
seconds were divided up. It impresses boys greatly that this 
sort of magnification or photographic enlargement might go 
on " for ever ". If, for instance, we take the first quarter- 
second of the last figure (2466), and magnify the distance 
line 16 times (as we did in the case of fig. 246a), we get still 
another replica, this time with the quarter-second divided 
up to show the happenings during each sixteenth-second. 
However short the distance, there is acceleration, and the 
acceleration has a constant value. The acceleration is 
" uniform ". 

" Uniform acceleration is measured by the amount by 
which the velocity increases in unit time." Many boys 
have difficulty in understanding what " uniform " accelera- 



Fig. 2466 



THE CALCULUS: SOME FUNDAMENTALS 429 




Fig. 247 



tion, such as acceleration due to gravity, really implies. 
" If only you would accelerate by adding on velocity in 
definite chunks at equal intervals, we could understand it." 

Let the boy have his definite chunks, at first, and utilize 
these for approaching the main idea. Go back to the graph. 

Suppose a train to move for 1 minute at a uniform velocity 
of 5 miles an hour; then to be suddenly accelerated to 10 
miles an hour and to travel for 1 minute 
at that velocity; then to be accelerated to 50 
15 miles an hour for a third minute; to 20 
miles an hour for a fourth minute; to 25 for 
a fifth; and to 30 for a sixth. How far would 
it have travelled altogether? A velocity- 
time graph shows this at once. The 
number of units of area under the graph is 
1 + 2 + 3 + 4 + 5 + 6-21, and this 
gives us the number of miles travelled. 

The dotted line passing through the top left-hand corners 
of the rectangles can easily be proved to be straight, and 
this evidently indicates some sort of uniformity in the motion. 
But the whole of the area under this line is not enclosed by 
the rectangles; there are 6 little triangles unaccounted for. 
How are these triangles to be explained? By the fact that 
really we have imagined an impossible thing, viz. that at 
certain times the train's speed was instantaneously increased 
5 miles an hour. 

Now although in practice we know that even in the very 
best trains acceleration is really brought about by sudden 
jerks, these jerks are virtually imperceptible, and it is there- 
fore not impossible to imagine an acceleration free from such 
sudden increases. It may be easily illustrated by running 
water: the following ingenious illustration we owe to Professor 
Nunn. 

Attached to my bath is a tap so beautifully made that 
by means of the graduated screw-head I can regulate the 
amount of water running in up to 8 gallons a minute. 



430 



CRAFTSMANSHIP IN MATHEMATICS 



I turn on the tap for one minute, the water running at 
the rate of 1 gallon a minute; in that time 1 gallon has been 
delivered. Then I turn the tap on further, to deliver water 
at the rate of 2 gallons a minute, and allow it to run for one 
minute; during this minute, 2 gallons have been delivered. 
Thus I continue for 8 minutes, 8 gallons running in during 
the eighth minute. The graph (fig. 248, i) shows the water 
run in during the successive minutes; the shaded rectangle, 
for instance, represents the amount of water (5 gallons) 



GALLS s 

per 
M.N. 



MINUTES 



23456 

MINUTES 



Fig. 248 



run in during the fifth minute. Total number of gallons 
delivered = 36. 

I now repeat the operation, but this time I turn the tap 
on every half minute, beginning by running in \ gall, a 
minute, and increasing by \ gall, each half minute. The 
first delivery will be J gall., the next \ gall., and so on, the 
last being 4 gall. But note (fig. 248, ii) that during the last 
half of the fifth minute, when 2| gall, were delivered, the rate 
of delivery was 5 gall, a minute; this column has the same 
height as the corresponding column in (i), but, of course, 
only half its area. The rate of flow during the half minute 
was the same, though only half the 5 gall, was actually 
delivered. The rate of flow during the last half of the eighth 
minute was 8 gall, a minute, though only 4 gall, were de- 
livered. Total number of gallons delivered = 34. 



THE CALCULUS: SOME FUNDAMENTALS 431 



I repeat again, this time allowing the water to be in- 
creased every J minute, beginning by running in J gall, 
a minute, and increasing by J gall, each J minute. The 
first delivery is thus yg- gall., the next J gall., the last f|- or 
2 gall. Note (fig. iii) that during the last | of the fifth minute, 
when 1J gall, were delivered, the rate of delivery was still 
5 gall, a minute; the column has the same height as the 
corresponding columns in the first two figures, but of course 
only J of the area of the column in the first figure. The 

8 
7 

G 



O 2 




I Z 



3456 

MINUTES 
(in) 



1234 

MINUTES 
(iv) 



676 



Fig. 249 



rate of flow in the column preceding HK was the same in all 
3 cases. Total number of gallons delivered (fig. 249, iii) = 33. 

I repeat the operation once more, this time turning on the 
tap gradually and continuously, in such a way that at the 
end of the first minute the water is running at the rate of 
1 gall, a minute, though only momentarily; and so on. 
At the end of the eighth minute I turn off, i.e. at the very 
moment when the rate of flow has reached 8 gallons a minute. 
The graph (fig. 249, iv) is now a straight line, and its area is 
|(8 X 8) or 32 units, the number of gallons delivered. 

Observe that, in all 4 figures, the rate of delivery at the 
end of any particular minute is the same, for instance at 
the end of the fifth minute, represented by HK; though in 



432 CRAFTSMANSHIP IN MATHEMATICS 

the last case, when the tap is gradually turned on, the rate 
at any particular time is only momentary, since the rate is 
continuously changing. 

In the last figure, HK no longer bounds a rectangle, as 
it did in the previous three figures; all the columns have 
become indefinitely narrow. The column which HK bounded 
has shrunk to a mere line which therefore cannot represent 
any quantity of water delivered. Still, as it has the same 
height as the series of gradually narrowing columns which 
it bounded, we say that it represents a rate of flow of 5 gall. 
a minute, just as the columns did. But this rate of flow 
is clearly not a rate of flow during any interval of time, 
however small. Hence we say it is the rate of flow at the 
end of the fifth minute. 

Boys ought now to understand clearly that the velocity 
of a body at any instant is measured by the rate per unit 
time in which distance is being traversed by the body when 
in the immediate neighbourhood of that instant. 

A body cannot move over any distance in no time, so that 
we could not find its velocity by observing its position at 
one single instant. To find its rate of motion, we must 
observe the distance traversed during some interval of time 
near the given instant, this interval of time being the shortest 
possible. Hence the term velocity at any instant must be 
regarded as an abbreviation for average velocity during a 
very small interval of time, including the given instant. But 
we have no means of finding such a velocity by actual experi- 
ment. We have to adopt other means. 

It is sometimes said that acceleration at a given instant 
of time is measured by the rate per unit time at which the 
velocity is increasing in the immediate neighbourhood of the 
given instant, or the average acceleration in a small interval 
of time including the given instant. 

The question, what is meant by the statement that at a 
certain moment a thing is moving at the rate of so many 
feet, a second ought now to be answered by all average 



THE CALCULUS: SOME FUNDAMENTALS 433 



pupils. Sixth Form boys should grasp the full significance 
of the following formal statement: " if the magnitude 
possessed by any increasing or decreasing quantity be re- 
presented by an area-function, the rate of increase or decrease 
of the quantity at any specified point is given by the corre- 
sponding ordinate function/' 

Thus if any given function is regarded as an area-function, 
the corresponding ordinate function may be called the 
rate-function of the former. 

Calculation of Rate -functions 

We may consider again the rate-function corresponding 
to the area-function ax 3 . According to the results at the 
end of the last chapter, this should be 3ax 2 . 




Q X 



(i) 



Fig. 250 



Let the curve in fig. 250 have the property that the area 
under it from the y axis up to any ordinate PQ is ax 3 . How 
may we determine the exact height of PQ? 

Take two other ordinates (fig. ii) CD, EF, each at distance 
h from PQ. Draw upon DQ, QF rectangles whose areas are 
respectively equal to those of the strips under the curve 
between CD and PQ, and PQ and EF. Let the curve cut 
the upper ends of these rectangles in p and p'. Draw the 
ordinates pq and p'q'. 

(E201 29 



434 



CRAFTSMANSHIP IN MATHEMATICS 



Although we cannot calculate PQ directly, it is easy to 
calculate pq and p'q 1 '. We have: 



pq X h = area CQ. 
. _ area CQ 
mpq h~ 

_ ax 9 a(x / 

h 

= (3* 2 - 3xh + 
i.e. /><? - {3* 2 - h(3x - 



X h = area EQ. 

..//__ area EQ 
.<, _ 

a(x + /O 3 



i.e. p'q' = {3* 2 + (3* + /*)} 



Whatever value ^ may have, h may be taken smaller; hence 
h must be smaller than 3x. Thus h(3x A) and A(3# + h) 
must both be positive, and pq will necessarily be less than 
PQ, and p'q' greater than PQ. By making h small enough, 
we can make pq and p'q' differ from Sax 2 as little as we please. 
In other words, PQ must lie between all possible positions 
of pq and p'q' y and thus the value 3ax 2 is the only value left 
for it to possess. 

The Rate as a Slope. Here is another way of consider- 
ing a rate-function. Let OQ' be the curve y = ax 3 . Let 
the abscissa of any point P be x, and the abscissae of two 
neighbouring points Q and Q', x h and x + h, respectively. 
While x increases from x - h to x y and from x to x + A, 
y increases by Q<? and q'Q', respectively. (Fig. 251.) 

Hence the average rate of the latter increases must be 

S*? and ?yi, i.e. tanP*X and tanPz'X, respectively. 



___ PM - QN 

~~ h 

_ ax* - a(x - Kf 

h 
i.e. tan P*X = {3* 2 - h(3x - h)}a. 



tan 



h 
Q'N' - PM 



- ax 3 



i.e. tan 



>? + h)}a, 



(both as in the last example) 
As h gets smaller, Q and Q' approach P, tanP/X being always 



THE CALCULUS: SOME FUNDAMENTALS 435 

less, and tanP^'X always greater, than Sax 2 , though by taking 
h small enough, they may be made to differ as little as we 
please from Sax 2 . 

If PT be drawn, so that tanPTX = 3a* 2 exactly, then 
PT is evidently the tangent at P. For a line through P ever 
so little divergent from PT would make with the x axis an 
angle greater or less than PTX, and so would cut the curve 




h, is purposely exagge rated 
ib make the -Kdure clear 



Fig. 251 

in one of the possible positions of Q or of Q'. Hence PT is 
the only line which meets the curve at P but does not cut it. 

Thus PT holds among secants such as PQ or PQ' the 
same unique position that HK holds among the rectangles 
(figs. 248, 249), or that PQ holds amongst the other ordinates 
(fig. 250). 

In fig. 251, the slopes of PQ and PQ' measure the average 
rate of change of the function during the changes of x repre- 
sented by NM and N'M. The slope of PT does not measure 
the change during any intervals, but evidently measures what 
has been defined as the rate of change of the function at the 
moment (or for the value of x) represented by OM. 



436 CRAFTSMANSHIP IN MATHEMATICS 

Meaning of " Limit " 

The common element in the three cases considered, 
HK (figs. 248, 249), PQ (fig. 250), PT (fig. 251), is described 
by saying that all three are examples of a limit. In all 
three cases, members of a series have been brought nearer 
and nearer the limit, but they have never been so near that 
they could not have been brought nearer. They have always 
remained " in the neighbourhood " of the limit, but in 
every case the limit has been unreachable. In all three cases, 
the limit is the first number outside the series. 

A rate-function is sometimes given this general definition: 
Take the given function of x, and find how much its value 
changes when x is raised or lowered by any positive number 
h. Divide this change by h, and so obtain the average rate 
of change for a change of the variable from x h to h y or 
from x to x + h. The rate-function is the limit of the quotient 
and is indicated more and more closely as h gets smaller 
and smaller. 

The Two Main Uses of Limits 

1. To define the velocity of a given point at a given moment. 
If we define velocity as the quotient of a distance travelled, 
by the time in which it is traversed, then " the velocity at 
a given moment " is not a velocity at all. 

On the other hand, if we consider the distance travelled 
by the point during a series of constantly decreasing in- 
tervals of time, and divide each distance by the length* of the 
corresponding interval, we shall again fail, as a rule, to 
obtain anything that can be called the velocity of the point, 
for^all the results will be different, except in the special case 
of uniform motion. But if the sequence of average velocities 
thus calculated follows some definite law of succession as 
the interval is taken smaller, then it will generally have a 
definite limit as the interval approaches zero. Thus the 
limit is a perfectly definite number, associated in a perfectly 



THE CALCULUS: SOME FUNDAMENTALS 437 

unambiguous way, both with the given moment and with 
the endless sequence of different average velocities. More- 
over, for small intervals of time, the average velocities are 
sensibly equal to the limit, the differences being of theoretical 
rather than of practical importance. It follows that although 
the " velocity at the given moment " is not really a velocity 
at all, it is quite the most useful number to quote in order 
to describe the behaviour of the moving point while it is 
in the neighbourhood of the place which it occupies at the 
given moment. 

2. To determine a magnitude which cannot be evaluated 
directly. Consider again fig. 250. We had to determine the 
height of the ordinate PQ. We found (i) that it lies between, 
and is the limit of, a lower sequence consisting of ordinates 
pq and an upper sequence consisting of ordinates p'q'\ (ii) 
that it lies similarly between, and is the limit of, the se- 
quences of numbers represented by { 3# 2 h(3x - h) } a 
and { 3# 2 -f- h(3x + h) } a\ and (iii), that the latter sequence 
corresponds to the former, term by term. From these 
premises it seems to be an inescapable conclusion that the 
height of PQ is exactly 3## 2 , for PQ is the only line between 
the two sequences of (i), and 3ax 2 is the only number between 
the two sequences of (ii). 

For blackboard revision work occasionally, devise questions 
to emphasize these principles (the term gradient might now 
be used generally): 

(1) The gradient of a chord is the average gradient of the 
arc. 

(2) A tangent is the limiting position of a secant. 

(3) The gradient of a tangent at P is the gradient of the 
curve at P. 

(4) The gradient of the tangent is the rate at which the 
function is changing. 

(5) The limiting value of the slope of a secant is the slope 
of the tangent. 

" In the neighbourhood of." We have spoken of the 



438 CRAFTSMANSHIP IN MATHEMATICS 

members of a series being " in the neighbourhood of " a 
limit. What is a neighbour! That is a question of degree. 
In Western Canada, a man's nearest neighbours might be 
40 or 50 miles away; in an English country district, perhaps 
a single mile; in a town, only a few yards; round one's own 
table, only a few inches. So with number sequences: it is 
just a question of degree. For instance, we know that TT comes 
within the interval 3-1 and 3-2, and therefore 3-1 and 3-2 
are neighbours of TT. But TT also comes within the smaller 
interval 3-13 and 3-15, which are therefore closer neigh- 
bours of 77. Again, TT comes within the interval 3*1414 
and 3*1416, which are therefore still closer neighbours of TT. 
And so we might go on. However close our selected neigh- 
bours of 77, we can always find still closer neighbours. Thus 
77 always has neighbours no matter how small the interval 
in which he is enclosed. It is all a question of standard of 
approximation. The important thing, when dealing with 
limits, is that we must never think of the interval shrinking 
to nothing. Think of the interval as always large enough for 
standing room both for 77 itself and some neighbours. The 
neighbours cannot be thought of as disappearing altogether. 

Secant to Tangent Again 

The gradient of a straight-line graph AB is determined 
easily enough: it is the ratio, ordinate j> /abscissa x. The 

ratio may be determined 
from any selected bit of 
the line. Or, if we like, 
we may increase the line, 
say to BC,and take the ratio 
CD ( increment of y) to 

BD ( increment of x). 

Fig . 2S2 But if the graph is a 

curve, the gradient at any 

specified point on the curve is determined by the tangent 
at that point. A ruler held against the edge of an ordinary 




THE CALCULUS: SOME FUNDAMENTALS 439 

dish is, practically, a tangent at a point on the ellipse. If, 
then, we want to determine the gradient of a curve, why 
not just draw the tangent and measure the angle it makes 
with the x axis? 

With a circle this would be easy enough: we should draw 
a radius to the point and then a line at right angles; and 
there are simple rules for certain other curves. But merely 
to hold a ruler against a curve, and to draw a line, is not to 
draw a tangent that we can accept. Circulate amongst the 
class copies of a mechanically drawn parabola, tell the boys 
to draw a tangent at the point P, and then to measure the 
angle that the tangent makes with the x axis. The angles 
will probably be all different. Clearly the method will not 
do, for the angles ought to be the same in all cases. 

If we draw a secant instead of a tangent, and find the 
gradient of the secant, we shall evidently have the average 
gradient of the curve between the two points P and Q where 
the secant intercepts the curve. Would that help? 

Yes, but if the points are 
far apart, as P and Q x , the 
slope of the secant, and 
therefore the average gra- 
dient of the curve between 
the two points, differs much 
from the gradient of the 
tangent PT. If we bring 
the points closer together, 
say P and Q 2 , the gradient 
of the secant is nearer the 
gradient of the tangent. If p . 

we bring Q down to Q 3 the 

gradient of the secant PQ X is still nearer the gradient of the 
tangent. It is this gradient of the tangent that we have to 
find somehow. 




[Some teachers prefer that numerical considerations like 
those that follow should precede the more general work 



440 



CRAFTSMANSHIP - IN MATHEMATICS 



concerning the graph, as outlined in the earlier part of this 
chapter. I have seen equally satisfactory final results obtained 
from both sequences.] 

Let us actually calculate the gradients of successive 
secants, and see if we can learn anything from the results. 
On a parabola we will select a point P where x = 1 (and 
.". y = I 2 1), and keep this fixed. We will also place a 
point Q on the curve, at first where x = 1-5 (and .'. y = (1*5) 2 
= 2-25). Thus, since P is (1, 1) and Q is (1-5, 2-25), the 
increment of x is '5, and the increment of y is 1'25. (The 
piece of line PQ may be looked upon as an " increase " 
of the line AP; hence the term " increment " may usefully 
be applied to the corresponding increases of x and y.) 

The gradient of the secant = = == 2*5. Now 

PV '5 



1-25 






THE CALCULUS: SOME FUNDAMENTALS 441 

bring Q gradually closer to P. Let the next four x values be 
1-4, 1*3, 1-2, 1-1; then the corresponding y values are (1*4) 2 , 
(1*3) 2 , (1-2) 2 , (1*1) 2 . The gradient calculations may be sum- 
marized thus (the x and y increments are often indicated by 
h and k t respectively): 



* = ON = 


1-5 


1-4 


1-3 


' 1-2 


1-1 


y = x * = ON 2 = NQ = 


2-25 


1-96 


1-69 


1-44 

44 
2 


1-21 


* = QV- 

h = PV = MN = 


1-25 
5 


96 
4 


69 
3 


21 
1 

2-1 


Gradient - J = ^ - = 

h MN 


2-5 


2-4 


2-3 


2-2 



Note how the value of the gradient has diminished from 
2-5 to 2-1. We cannot write h = 0, or the denominator of 
our ratio would equal 0, and the ratio would have no 
meaning. But we may continue to diminish the values of 
the x increment, and calculate the gradient as before. We 
may make the increments as small as we please. Let us 
calculate the gradient when the successive values of x for Q 
are 1-01, 1-001, 1-0001, 1-00001, so that the x increments 
are -01, -001, -0001, -00001. We cannot draw the figure, 
for the increments are much too small to be shown. 



* = ON 


1-01 


1-001 


1-0001 


1-00001 


y = x 2 = ON 2 = NQ = 


1-0201 


1-002001 


1-00020001 


1-0000200001 


fc = QV = 

h = PV - MN = 


0201 
01 


002001 
001 


00020001 
0001 


0000200001 
00001 


Gradient - \ = J - = 

h MN 


2-01 


2-001 


2-0001 


2-00001 



442 CRAFTSMANSHIP IN MATHEMATICS 

We observe (1) that however small we make h (the in- 
crement of x), the value of the gradient always exceeds 2; 
(2) that the smaller we make the increment, the smaller is 
the excess of the gradient over 2. Evidently we can approach 
to within any degree of approximation we like to name; it 
is only a question of making h small enough to start with. 
We observe also that the more nearly the value of the gradient 
approaches 2, the more nearly does the secant approach 
the position of the tangent. As long as the secant remains 
a secant, it can never be a tangent, and it must always have 
a gradient in excess of 2. But the successive gradients seem 
to compel us to infer that the gradient of the tangent itself, 
and therefore of the curve, is 2. Thus we regard 2 as the 
limiting value of the gradient of all possible secants. It is a 
value that is never quite reached by any secant, for the tangent 
stands alone, outside them all, four-square and defiant! 

Thus we have found that, for the function y = x 2 , the 
gradient of the point P, where x 1, is 2. 

We may arrive at the same result by arguing more generally, 
merely calling the increments, h and k. 

The co-ordinates of P are (1, 1). 

The co-ordinates of Q are (1 + h> 1 + &) 

Since y = x 2 , 

(1 + *) = (1 + h? 



:. k = 2h + h 2 , 

QV k 2h 
---- 



From this point on, argument nowadays commonly 
proceeds thus: 

As Q approaches P, so h tends towards 0. We have to 

k 2/z -4- h 2 

find the limit to which - or '- tends as h tends towards 
h h 

9/j _1_ /2 

As long as h is 4= 0, Ln \ n = 2 + h, and as h-+ 



THE CALCULUS: SOME FUNDAMENTALS 443 

2 + h -+ 2. If we decide that 2 + h must differ from 2 by 
less than 1/1000000, there is no difficulty; we merely give 
to h a value less than that, e.g. 1/1000001. 

k 

In the limit, as h -> 0, T -> 2. 
A 

Hence the gradient of the curve at P = 2. 

We may now find the gradient at any point P (x y y). 
The co-ordinates of Q are (x + A, j> + A). 

.' (y + *) = (* + A) 2 

= * 2 + 2xh + A 2 , 

/. k =-- 2xh + A 2 . 

Hence the gradient of PQ = ^ = 2x + h if h 4= 0. 

A 

Now as Q approaches P, A -> 0. 
/. the gradient of the curve at P limit of (2x-\-h) as A-> 0, 



I am not quite happy about the language of this argument, 
though it is now in common use and has been designed to 
get over the old difficulty of infinitesimals and of the absurdity 
of dividing by 0. But even able boys in the Sixth sometimes 
admit that the reasoning is not clear to them, saying that 
they feel they take a leap over the final gap to the limit. 
The teacher must insist that the gap is really never crossed, 
that the interval still remains, that the limit is always there 
with a crowd of neighbours who vainly strive to reach him; 
that every neighbour has a value a little greater than 2x (or, 
in some of our earlier illustrations, a little less), and that the 
value 2x is a solitary value, which therefore we feel bound to 
assume is the value which belongs to the Limit, and to the 
Limit alone. 

Revise: The function y = x 2 . To calculate the ordinate 
for any value of x, work out the value of x 2 . To calculate the 
gradient for any value of x, work out the value of 2x. 



444 



CRAFTSMANSHIP IN MATHEMATICS 



Thus x 2 may be described as the formula for the ordinate, 
and 2x as the formula for the gradient. In other words, the 
function x 2 gives the ordinate, and the function 2x the gradient, 
for any value of x. 

x 2 is the original function which defines the curve; 2x 
is called the derived function of x 2 . The process of finding 
the derived function of a given function is called differentiation. 

Since the gradient of the tangent to y x 2 at any point 
P is 2#, the gradient where x = 1, is 2; where x = 2, is 4; 
where x = 3, is 6; &c. Does this square with the work we 
have done in pure geometry? We found (p. 409), 

j-^r^ ^ r u 1 i latus rectum 

gradient of tangent to axis of parabola = 
5 * F 



or gradient of tangent to tangent at vertex = 



ordinate 
ordinate 
latus rectum' 



Let the tangent at the vertex be the x axis, and let the axis 




of the parabola be the y axis. Let S be the focus, and let 
the latus rectum LSP be unit length. 



THE CALCULUS: SOME FUNDAMENTALS 445 

Half the latus rectum = SP = |. Since PN = half the 
distance of P from the directrix (not shown), PN = |PS = J. 
Hence the co-ordinates of P are (, ). 

At the point Q (1, 1), gradient of tangent to OX 
ordinate __ 1 _ ~ 



half latus rectum 

At the point W (2, 4), gradient of tangent to OX 
ordinate _ 2 __ . 



half latus rectum \ 

At a point Z (3, 9), gradient of tangent to OX 

ordinate 3 , 

= - = o. 



half latus rectum \ 

Clearly then, the new method of finding the slope of the 
tangent does produce a result absolutely accurate, not merely 
approximately accurate. Evidently the " limit " argument is 
sound, though we must always remember that the limit 
is outside the sequence under consideration, never reached 
by any member of the sequence. 

The Calculus Notation 

We have used the letters h and k to denote the increases 
(" increments ") in the values of x and y. But the increments 
always actually considered are very small, and the symbol 
generally used to denote them is the Greek letter delta 
(A or 8) prefixed to the value of x or y from which the in- 
crement begins. Pronounce A* as " delta x "; the symbol 
A# must be taken as a whole; A is not a multiplier and has 
no meaning apart from the x and y to which it refers. 
Remember, then, to write Ax instead of /r, and by instead 
of*. 



446 CRAFTSMANSHIP IN MATHEMATICS 

A# means " the increment of x "; Ay means " the in- 
crement of y ". 

Ay - . increment of y rp, A , , 

means the ratio -=- . 1 he A s cannot be 

A# increment or x 

cancelled. 

Treat - exactly as if written -; it measures the average 

A# h 

gradient of the graph over the interval between x and 

x + A#. 

The limit of ^ is the gradient of the graph at the point 

. dy 

given by x. It is sometimes written D(y), sometimes -^-. 

j ax 

But the curious thing is that, although ~ looks like a ratio 

dx 

or a fraction, it is not a ratio or fraction. The symbols dy, 
dx, written separately, have no meaning. The limit of A# 

is not dx\ the limit of Ay is not dy. ~- is just a single symbol. 

dx 

-~ is always a ratio of real value; -f- is not a ratio at all 
A# dx 

and is therefore very misleading to the eye. 

The process of finding D(y) or is called differentiation. 
, ax 

D(y) or -f- has received various names: 
dx 

(1) The derivative of y or f(x) with respect to x. 

(2) The differential coefficient of y with respect to x. 

(3) The derived function of y with respect to x. 

We will differentiate x*. Let y = re 4 . When x is increased 
to x + A#, let jy be increased to y + Ay. Then: 

y + Ay = (x + A*) 4 

= * 4 -f 4* 3 A* -f 6* 2 (A*) a + 4*(A*) 8 + (A*) 4 ; 
-f 6* 2 (A*) a 4- 4*(A*) 3 -f (A*) 4 , 



THE CALCULUS: SOME FUNDAMENTALS 447 

Hence as A* -> 0, - ~- -> ky?. 

A* 

.'. ^ = 4*'. 
dx 

Let the class discuss the result (or one like it) critically. In 
particular, discuss the significance of the arrows. Forms 
and language that might pass muster in an examination room 
should be subjected to the closest scrutiny in class. It is a 
fact that those boys who have acquired facility in working 
out the ordinary stock exercises in the calculus are often 
nonplussed when cross-examined in the underlying funda- 
mental notions. 

There can be no doubt that the idea of derived functions 
is best introduced as a generalization of the familiar 
ideas of connexions between area functions and ordinate 
functions, ordinate functions and gradient functions, &c. 

The notation -f should not be introduced too soon. D(y) 
ax j 

is much preferable. The symbol ~- originated with Leibniz 

(not with Newton), and it expresses a view of the nature of a 
differential coefficient that is out of harmony with modern 
ideas and conflicts with the doctrine of limits. Originally 
the view was that any finite value of the variables y and x 
is really the sum of a vast number of " infinitesimal " values 
which, though immeasurably small, have yet a definite 

magnitude. Thus the differential coefficient -f- was looked 

dx 

upon as simply the ratio of the " infinitesimals " of 
two variables, the ratio being finite and measurable (much as 
the weights of atoms are measurable), in spite of the smallness 
of the terms. This view is no longer held. The expression 

-^ is not a ratio at all but only the limit which the ratio 
dx 

of the increases of the variables approaches as the increment 
of x approaches zero. Naturally the learner is greatly puzzled 



4:48 CRAFTSMANSHIP IN MATHEMATICS 

if he is told to write the derivative in the form of a 
fraction and is then forbidden to think of it as a fraction. 
Thus it is much the best plan to withhold the Leibnizian 
notation at first. Use the symbol D(y) instead; this symbol 
reminds the pupil that he is seeking a function which he is 
to derive from the given function y by means of a definite 
rule of procedure. This relationship between functions is 
the essence of the whole matter. 



Integration. Like current ideas about the nature of a 
differential coefficient, those about the nature of an integral 
also show traces of the erroneous mathematical philosophy 
of earlier days. The problems first systematically studied by 
Wallis came to be regarded as having for their aim the 
summation of an " infinite " number of " infinitesimals " dy, 
of the form y.dx. This view is still represented not only by 
the usual notation I = \y.dx, which (like dy/dx) was in- 
troduced by Leibniz, but also by the common statement 
that an integral is the sum of an infinite number of infinitely 
small magnitudes. With the rejection of the notion of an 
infinitesimal as a definite atomic magnitude, this statement, 
and the notation which expresses it, have become inadmissible. 
If dx has any numerical significance at all, it stands for the 
increment h when h is zero. Hence the product y.dx is also 
zero for all values of y, and therefore the summation repre- 
sented by jy . dx is the summation of a series of zeros! I is 
not the sum of an infinite number of products; * it is 
simply the limit of the sum of a finite number of pro- 
ducts. 

There is neither need nor warrant for introducing the 
term " infinite " at any point of the discussion. If we sub- 
stitute the useful A# for the absurd dx y we may still usefully 
retain the Leibnizian mode of expression I = fy . A#, but the 
symbol " / " must now be read, " limit of the sum as A# 
approaches zero ". 



THE CALCULUS: SOME FUNDAMENTALS 449 
Interpretation of -. 

n Ay distance j j A 

1. -r^- = : = average speed during A#; 

A* time * F fe 

dy limit of average speed = " instantaneous " 

d~x ~~ speed. 

2. - average slope of curve during interval A#; 

L\X 

j- = limit of average slope = limit at point P. 

The two problems (1) to determine the rate of increase 
of a function and (2) to draw a tangent to a curve, are really 
identical; if we have a general method of determining the 
rate of increase of a function f(x) of a variable x, we are 
able to determine the slope of the tangent at any point (x, y) 
on the curve. 

Points for emphasis. We will once more stress the points 
to be kept in the forefront of the teaching. 

The pupils should be told plainly that the old idea of 
infinitely small quantities has been definitely abandoned. 
The real explanation of the whole thing was first put forward 
by a German mathematician, Weierstrass, about the middle 
of the nineteenth century. The subject had been sound 
enough; so, virtually, had been the mathematical procedure, 
but the explanation had been wrong. 

The problem was to retain an interval of length A, over 
which to calculate the average increase, and at the same time 
to treat h as if it were zero. As Professor Whitehead says, 
" As long as we look upon ' h tending to a ' as a fundamental 
idea, we are in the clutches of the infinitely small, for we 
imply the notion of h being infinitely near to a. This is 
what we want to get rid of." " The limit of f(h) at a is a 
property of the neighbourhood of a." " In finding the limit 

(E291) 30 



450 CRAFTSMANSHIP IN MATHEMATICS 

Z/Ov I Jj\ 

of -^ 1 at the value of the argument h y the value 

h 

(if any) of the function at h = is excluded. But for all values 
of h except h = we can divide through by h." " In the 
neighbourhood of the value for h, 2x + h approximates 
to 2x within every standard of approximation, and there- 
fore 2x is the limit of 2x + h at h = 0. Hence, at the value 

for h, 2x is the limit of (* + h Y ~ ** 

h 

The difficulty of former mathematicians was that on the 
one hand they had to use an interval h over which to calculate 
the average increase, and on the other hand they wanted to 
put h = 0. " Thus they seemed to land themselves with 
the notion of an existent interval of zero size." Present-day 
mathematicians avoid that difficulty by using the notion 
that, corresponding to any and every possible standard 
of approximation, there is still some interval. 

My own experience is that when Sixth Form boys are 
puzzled over this question, their puzzlement is almost always 
due to the fact that they have got hold of the term infinity, 
and do not understand what the term signifies. 

Books to consult: 

1. The Teaching of Algebra , Nunn. 

2. An Elementary Treatise on the Calculus, Gibson. 

3. Course of Pure Mathematics, Hardy. 

4. Applied Calculus, Bisacre. (An outstanding book.) 



WAVE MOTION 461 

CHAPTER XXXI 

Wave Motion: Harmonic Analysis: 
Towards Fourier 

Sine and Cosine Curves. Composition 

The pupils will, of course, be thoroughly familiar by 
this time with the radian notation, and will understand that 
the reason for measuring angles in radians is that theoretical 
arguments are simplified. They will know that TT radians= 180; 
that as an angle of 9 radians is subtended by an arc of Or, 
the length of an arc of a circle = r9\ that the symbols 6 and 
(/> are commonly used in circular measure, and the symbols 
a, /?, y for measurements in degrees. They ought also to 
know that, when an angle is small, its circular measure may, 
in approximation calculations, be substituted for its sine 
(or tangent); and that, when it is not small, the values of 
the sine and cosine may still be expressed approximately in 
circular measure by means of the simple formulae sin 9 = 

3 9 2 

, cos0 = 1 . The proofs of these may be given at 
6 2 

an appropriate stage, but a simple graphic method is easily 
devised to suggest that the formulae are approximately true. 

When boys are first introduced to angles greater than 
360, they are inclined to doubt if they are dealing with real 
things, and to be a little sceptical about the practical value 
of the work in hand. Light, however, begins to dawn when 
they are introduced to Simple Harmonic Motion, to waves, 
and to spirals. 

They must be made to understand clearly that the values 
of the ratios connected with an angle are repeated endlessly 
in cycles as the angle rises. 

They will, of course, be thoroughly familiar with the sine 
and cosine curves. With very little practice they can make 



452 



CRAFTSMANSHIP IN MATHEMATICS 



a supply of these curves for themselves by running them 
off from Fletcher's trolley arranged for uniform motion; 
with care, these curves may be obtained to a surprising 
degree of accuracy. Draw tangents to the succession of 
crests, then the axis midway between them. The chief 
ordinates are the perpendiculars at those points of the axis 
midway between the nodes. 

Point out that all sine curves have the same general 
shape and properties; that smx gives a wave curve of period 

Y 



2?r with successive maximum and minimum values at +1 

and 1 respectively; that sinpx gives a wave curve of period 
o o 

; that asin(px-\-e) gives a wave curve of period , 
P P 

with successive maximum and minimum values of -\-a and 
a, respectively, the effect of e being merely to displace 
the curve along the axis. These fundamentals must be 
mastered. The p y the a y and the e are veritable traps for 
the unwary beginner; the inner significance of the three 
symbols should be expounded and emphasized again and 
again. 

From his earlier knowledge of graphs, a boy may, without 
further instruction, graph one or two easy cases of compound 
periodic functions. We give two examples adapted from 
examples in Siddons and Hughes' Trigonometry, the first, 
2 sin* + 3 cos#, consisting of two periodic functions of 
the same period (fig. 256), and the second sin3# + 2 sin*, 
periodic functions of different periods (fig. 257). In each 
case the two functions are first plotted separately (the curves 
are shown by lighter lines), then the required composite 



WAVE MOTION 



453 



curve is obtained by means of points determined by taking 
the algebraic sum of the ordinates of the constituent curves. 

Y 

4 




1'ig. 256 



For instance, in the second case pm = pn + pq. Note in 
the first case where we are compounding functions of the 
Y 






\ 



^ 

Fig. 257 



period, the result is a 'e curve; in the second case, 
where the functions to be compounded are not of the same 



454 CRAFTSMANSHIP IN MATHEMATICS 

period, the result, though a periodic curve, is not a sine 
curve. If in the second case we slide the half curve TT to 2ir 
along the x axis up to the y axis, the upper and lower halves 
will easily be seen to be symmetrical. Observe that in the 



case of sin 3x the period is - = 120. 

<j 

In all such cases the shape of tlje composite curve can 
be obtained only by plotting a wide range of ordinate values, 
though this is always simply done by algebraic addition, 
and a pair of dividers will soon give the necessary number 
of points. The curves are a little tricky to draw, because 
of the minus quantities to be added. 

This kind of exercise need take but little time. I have 
known boys work through half a dozen in an hour. The 
general shapes of the sine and cosine curves are already 
familiar, and as the axis can be divided up and the principal 
ordinate put in at once, the constituent curves can be sketched 
in in less than a minute. It is assumed, of course, that the 
significance of p, a, and e has been fully grasped. But the 
negative quantities to be considered when building up the 
composite curve nearly always give trouble. No calculations 
are, however, necessary. Let the dividers do the work, 
unless, in some very exceptional instances, rigorous accuracy 
is wanted. The general form and what it teaches is the 
main thing. 

Waves and their Production 

Since, in most instances, waves are periodic phenomena, 
they afford excellent concrete examples of periodic functions. 
No one can appreciate the most striking triumphs of physical 
science who has not given some attention to the mathematics 
of wave motion. In fact, wave motion now forms the very 
basis of the study of the greater part of physics, and, after 
all, the necessary mathematics of the subject is, in all its 
main factors, quite simple. It need hardly be said that 
that new and rather formidable subject, Wave Mechanics, 
is outside the scope of school practice. 



WAVE MOTION 455 

The teaching of such characteristics of waves as resistance, 
persistence, and over-shooting the mark, is part of the business 
of the physics master. The mathematical master is concerned 
mainly with considerations of the form of the wave and its 
analysis. Let beginners first read through Fleming's Waves 
and Ripples, and so supplement the work they have already 
done in the physics laboratory; the mathematics will then 
give them little trouble. But, if they begin the mathematics 
of wave motion before they have acquired in the laboratory 
a considerable amount of practical knowledge of the subject, 
they will never be quite sure of their ground. 

The boys will probably be familiar with the device of 
producing a train of waves by means of a length of narrow 
stair carpet, or a sand-filled length of rubber tubing, or a 
length of heavy rope: these things are part of the stock 
in trade for teaching wave motion in the physics laboratory. 
An instructive experiment is the following: take a common 
blind-roller about 5' long, with a pulley runner fixed at each 
end. Into the roller drive 37 4" nails, at 1J" intervals, in 
the form of a uniform spiral of 3 complete turns. The nails 
should be separated from one another by a uniform interval 
of 30, so that the 1st, 13th, 25th, and 37th are in the same 
straight line; the 2nd, 14th, and 26th in another straight 
line; and so on. Support the roller in a horizontal position 
in front of a white screen, and turn it by means of an im- 
provised crank. Let a distant light throw on the screen 
a shadow of the rotating roller. Observe how the shadows 
of the nail-heads exhibit progressive wave motion. Observe 
the movement of any one particular shadow; it is an example 
of simple harmonic motion (see Chapter XXXVII). The 
travelling shadow-wave, constituted by equal simple harmonic 
motions of the shadows of the nail-heads, is a progressive 
harmonic wave. The shadow of any head differs in phase 
from that of its neighbours by a constant amount of 30. 
Note that each nail remains in its own vertical plane; the 
progressive horizontal movement is one of form only. The 
boys must distinguish between (1) the actual to and fro 



456 



CRAFTSMANSHIP IN MATHEMATICS 



movements of elements in a wave-medium, and (2) the move- 
ment of the wave itself. The second is merely an appearance, 
resulting from the successive movements of the first. The 
first has the effect of making successive sections of the medium 
(as we may conveniently call it) assume one after another 
the same shape. The shape therefore appears to be some- 
thing moving along. 

The waves on the surface of the sea, away from the shore, 
are good examples of progressive waves. If their outline 
were exactly a sine curve, as theoretically it should be, we 
should have an example of a harmonic progressive wave. 



Common Wave Formulae 



In figure 258, 



the wave-length L t L 2 = L 2 L 3 = A, 
the amplitude = PQ = a. 




I x 



Fig. 258 

If T is the periodic time of the wave (i.e. time to complete 
a vibration), it follows from first principles that v = A/T = A, 
where v = velocity and n frequency. 

Let Lj# be d\ let pq be h. Since the curve is a sine curve, 

A = 360 = 2?r. Hence the number of degrees in d = X d. 

277 A 

For -T- write p. Then, wherever q is taken along L^Lo, it 

A 

may be found, by actual measurement, that h = a sind. 
This is a fundamental formula. 

Let y = a sinpx describe the wave outline in a given 
position, x being measured from Lj. If the curve move to the 



WAVE MOTION 457 

right with a velocity v, its form after t seconds is given by 
the formula, 

y a s'mp(x vt) (i) 

If to the left with the same velocity, 

y = a sinp(x + vt) (ii) 

These formulae are simply applications of the general prin- 
ciple that if a graph is moved a distance d parallel to the 
x axis, (x d) must be substituted for x in the formula. 

Since p , (i) and (ii) may be expressed thus: 
A 

y = a sin (x vt). 

A 

The actual significance, in the graph, of each symbol in 
this formula must be understood. 

Compound Harmonic Waves 

Let two boys near each other on the edge of a pond, 
or other suitable sheet of still water, each produce a series 
of waves by striking the water rhythmically with a stick. 
Let the frequency of the blows be 2 to 3 (say 2 in 2 seconds 
and 3 in 2 seconds, easily done after a little practice with 
watch in hand), and suppose the waves to travel with the 
velocity v. A pattern will result from the five waves which 
are produced every two seconds, and it will be regularly 
repeated, though gradually fading away into ripples. But 
this pattern will no longer represent simple harmonic waves, 
for the shape which appears to move along the water beyond 
the ends of the line joining the centres of disturbance, is 
no longer a simple sine wave; the length, the frequency, 
and the amplitude of the resultant waves will be different 
from the length, the frequency, and the amplitude of the 
component waves. At points reached simultaneously by 
crests and troughs belonging to the component wave-trains, 



458 CRAFTSMANSHIP IN MATHEMATICS 

the elevation or depression of the surface is exaggerated. 
All this should be confirmed by observation. 

To calculate the resultant disturbance due to the two 
component waves (assumed to be simple harmonic waves), 
we adopt the principle, which accords with observation, 
that the actual displacement at any point is equal to the 
algebraic sum of the displacements due to the waves separately. 

If the first wave-train existed alone, the displacements 
produced would be represented by moving the curve 

y = a : sin x with velocity v towards the right. If the 

^i 
second wave-train existed alone, the displacement produced 

would be represented by moving the curve y~a 2 s'm- (x c) 

2 
with velocity v towards the right. Here, c is the x co-ordinate 

(at t = 0) of the nearest point of the wave from the centre 
of disturbance, comparable with L in fig. 258. 

Thus the actual character of the resultant composite 
wave is represented by the graph 

. 2ru , . 27U, x 
y = a l sin x -j- a 2 sin (x c), 
AJ X 2 

moving to the right with a velocity v. 

This evaluation from first principles is really very simple, 
but, unless it is associated with at least a little experimental 
work, it may prove difficult for average boys. The formula 
is a key formula and should be mastered. 

Had there been three boys at the pond side, each pro- 
ducing waves by striking the water rhythmically, all the 
waves being of different length, the composite waves would 
have been more complex, and the necessary formula for the 
graph would have consisted of three terms. So generally. 



WAVE MOTION 459 

Comparison of Periodic and non -Periodic 
Functions 

Functions of the form s'mpx and cospx (where p = ~\ 

\ A / 

have much the same relation to periodic curves as x has to 
non-periodic curves. The simplest non-periodic curve is 
the straight line y Ax (we write it in different forms, 
according to circumstances; e.g. y mx -f- c); and the 
simplest periodic curve is y = A s'mpx (also written in 
different forms according to circumstances). 

With our former non-periodic work we soon learnt that 
the curve y = A I( * +A 2 # 2 was more complex than y = Ax, 
and that y A^x + A 2 x 2 + A 3 x? was more complex still; 
and so on; a quadratic function was more complex than 
a straight-line function, a cubic more complex than a 
quadratic. Still, however complex the function, it was 
always a question of the addition of a number of terms; 
the actual graphing was simple enough though tedious if 
the terms were many. 

So it is with periodic functions, where the curve, whether 
simple or complex, recurs endlessly, and makes a continuous 
wave. The effect of adding to y = A s'mpx the term A 2 sin 2px 
may be compared with that of adding to y A* the term 
A 2 # 2 ; in each case we obtain a form of greater complexity. 
By adding further terms we get, in each case, still further 
complexity, save that in the former case the successive curves 

all have the period A or a submultiple of this. 

P 
The standard form of a periodic function may be written: 

y Aj s'mpx + A 2 s'mSpx + A 3 sin3j># + . . . -f A r s'mrpx. 

It was the French mathematician Fourier who first observed 
that a periodic function of unlimited complexity may be 
described by a formula of this type. The process of deter- 
mining the components of which a given periodic function 



460 



CRAFTSMANSHIP IN MATHEMATICS 



is the resultant is known as harmonic analysis. Fourier's 
statement is known as Fourier's theorem. 

Let the boys consider an illustration of this kind. Let 
them inagine a water wave sent out with a velocity v, of 
length A, and frequency 1 per second; the wave would form 
a simple sine curve, such as we see on any disturbed water 
surface. Now let them imagine a second wave, sent out from 
the same point, independently but at the same moment, at 
a frequency of 2 per second, with the same velocity v and 
therefore of a wave-length A/2. This second wave would 
not have the appearance of its independent self but would 
be imposed on the other, and what we should see travelling 
along the water surface would be a composite wave. Now 
let them imagine a third wave to be sent out from the same 
point, independently but at the same moment as the other 
two, at a frequency of 3 per second, with the same velocity 
v, and therefore of a wave-length A/3. (Remember that 
v = nX y always.) This third wave will not, any more than 
the second, show itself independently; it will simply make 
the previous composite wave still more complex. And so 
we might go on. The waves sent out independently might 
be shown thus: 








[-, \ ( 



/ 





ot 


is 








/ 




s 






M 


s 




s 














S 




/ 








s 




/ 


Zir 






k 




/ 


x i - 





/ 


"\ 






/ 


\ 






/ 


\ 




M 


/ 


\ 






/ 


s 






s 


2. 






\ 


/ 






\ 


/ 


Zir 




\ 


/ 






-X 





Fig. 259 



WAVE MOTION 



461 



All 3 waves start from O, and since they travel with the 
same velocity they reach M at the same moment, but by 
that time the first will have completed 1 of its periods, the 
second 2, the third 3. Imagine a whole series of waves sent 
out in this way, each of them with a wave-length which is 
a submultiple of A, though not necessarily all the members 
of the sequence A, A/2, A/3, A/4, &c.: some of the series may 
be missing. Now imagine the water to be suddenly frozen, 
so that the wave would be set in ice, and its section readily 
drawn. We might have a composite wave like fig. 260, 
OM composing a unit which would be repeated endlessly 
until the wave died away in a ripple. The problem is, how 



/ 




\ 






^\ 




/ 




\ 




r\ . 








V 


y 




\^ 


/ 




\ 


J 


\J 


^ x ^ 



Fig. 260 

are we to analyse this curve, in order to discover all the 
simple curves of which it is compounded. 

The waves need not all have been sent out with the same 
amplitude (<z), as shown in fig. 259. Neither need they have 
been sent out from exactly the same point; one might have 
been started 40 or 50 farther along the axis than the others. 
And remember that a cosine curve is produced from a sine 
curve merely by pushing it forwards or backwards 90 along 
the axis. We may therefore easily find cosines as well as 
sines in our formulae; it is all a question of convenience, 
depending on the particular curves under consideration. 

Briefly, Fourier's statement was this: Any repeated 
complex wave pattern of length A may be produced by adding 
to a certain fundamental sine or cosine curve of length A, 
sine or cosine curves of the proper amplitudes whose lengths 
are A/2, A/3, A/4, &c. Conversely, the complex pattern may 
be revolved into its original component sine and cosine 



462 CRAFTSMANSHIP IN MATHEMATICS 

curves, since any of the unknown amplitudes may be deter- 
mined at will. 

Observe that the main difficulty in analysing the complex 
curve arises from the fact that the component curve may be 
of different amplitudes. The general expression for the 
complex curve may be written in different ways, though 
they all mean the same thing. 

(i) y = a + (#1 sin px + b l cos px) 

+ (a 2 sin2/># + b 2 cos2/>#) + . . . 
(ii) y = a + a sin(* + i) + # 2 sin(2# + a 2 ) 

+ a 3 sin(3# + a 3 )+ . . . 
(iii) y = a + a sin(0 + a) + b sin(20 + j3) 
+c sin(30 



Remember that p = 2?7/A. 

The constant a meets the case in which the x axis is 
not identical with the common axis of the various harmonic 
curves. 

Observe that each term in the above expressions represents 
a simple harmonic function. Those harmonics in which the 
coefficient of x is an odd number are called odd harmonics; 
those in which the coefficient of x is even are called even 
harmonics. Observe, too, that the second term gives a curve 
with twice as many complete waves, the third term a curve 
with three times as many complete waves (and so on), as the 
first or fundamental term. This is exemplified in fig. 259, 
where the " period " of the second term is \ the period of 
the first, the period of the third is \ the period of the first; 
and so on. The frequencies are therefore twice, three' times, 
&c., the frequency of the first. 

Curve Composition 

Let us first consider curve composition. It is very simple- 
Plot to the same axis the successive components, a l sin(#+ai) 
a z sin(2# + a 2 ), &c., and then add the corresponding ordinates 
to obtain the respective ordinates of the composite curve 



WAVE MOTION 



463 



(cf. figs. 256, 257). Since the first or fundamental term is 
represented by the period to 2?r, the wave will consist of 
repetitions of the first portion between x = and x = 360. 
We give the graph of 

100 sin* + 50 sin(3* 40), 

from to 360. The first or fundamental term, 100 sin x, 
represents the first harmonic with an amplitude 100 (=a l in 



100 



50. 




Fig. 261 

general formula). The second term, 50 sin(3# 40), is 
the third harmonic with an amplitude of 50 (=a 2 ), and 
consists of three complete waves within the period of the 
first. 

The function consists of only odd harmonics, and in 
virtue of this fact the graph possesses a special kind of 
symmetry characteristic of all curves containing only odd 
harmonics. If the portion of the graph from n to 2?r be made 



464 



CRAFTSMANSHIP IN MATHEMATICS 



to slide to the left, to the position to TT, it will be the re- 
flected image of the half above the axis. (Cf. fig. 257.) Note 
that the composite curve is not a sine curve. 

Had the function contained the absolute term a , say 

y = 70 + 100 sin x + 50 sin(3* - 40), 

the graph would be the same as before but raised vertically 
70 units. The line of symmetry referred to above would then 
no longer be the x axis. 

We give a second example, this time consisting of the 
first and second harmonics: 

y = 10 roi(6 + 30) + 5 sin(20 + 45). 

We will plot the graph from a tabulated series of values, 
though this is really unnecessary. 

If 10sin( + 30)= < y 1 , and 5 sin (20 + 45) = y 2 , then 



when = 0, yi = 10 sin 30 = 5, and y 2 = 5 sin 45 = 

i 

also when 6 = 30, y l = 10 sin 60 = 8-66 and y 2 = 5 sin 105 
= 5 sin 75 = 4-8. Similarly other values may be calculated. 




Note the device of running off horizontals from a graduated 
circle. Since the " period " in the x axis is divided into 12 
equal parts, we divide the circumference of the circle also 
into 12 equal parts. 



WAVE MOTION 



465 



Values ^ 
of 6 f 


0. 


30. 


60. 


90. 


120. 


y\ = 


5 


8-65 


10 


8-65 


6 


y2 = 


3-5 


4-8 


1-3 


-3-5 


-4-8 


yi + y* = 


8-5 


13-45 


11-3 


5-15 


0-2 



Observe that as the point on the smaller circle rotates at 
twice the rate of a point on the larger, it is only necessary to 
divide the smaller circle into half as many parts as the larger. 
Set up, say, 12 ordinates for the whole line to 360, then 
divide the circumference of the larger circle also into 12 
parts, and run off parallels to cut the ordinates. Each second 
harmonic will embrace only 6 of the 12 ordinates, and hence 
only 6 parallels from the smaller circle are required. The 
radii of the circles are, of course, equal to the amplitudes 
of the respective harmonics. Observe the plan for fixing 
the first point of each harmonic. 

Functions with more terms than two are treated in exactly 
the same way, but naturally the composition is a tedious 
operation. 

Curve Analysis 

Secondly, we come to the decomposition or analysis of 
a composite curve. This is much less simple than the reverse 
operation. 

The composite curve may be the resultant of two or more, 
perhaps a large number, of harmonics. But it does not at 
all follow that, because a particular harmonic, say the ninth, 
has been included in the building up, therefore all the earlier 
ones (in this case the first 8) of the series are included too. 
How are we to discover which harmonics are included, and 
how are we to draw them? 

Whatever scheme we adopt, it is advisable, when we have 
discovered the component harmonics, to draw them all 
carefully, to compound them again, and to see if the result 
corresponds to the original curve. 

(291) 31 



466 



CRAFTSMANSHIP IN MATHEMATICS 



Let integration wait until a later stage. Let the boys 
first learn what the new thing is really about. Let them 
consider the few simple cases which may easily be solved 
graphically, and after all, these are the cases of greatest 
practical importance (for instance those that are concerned 
with the theory of alternating currents). Such cases may, 
with sufficient approximation, be represented by the sum of 
two or three harmonic terms. 

We select as an example one of Mr. Frank Castle's 
engineering problems. The curve in the figure is drawn 



'% 

^ 

1* 

I 

vz 




j^" ^^Sk 






_jr IV 


7 [\ 




O 1 2 3 4\ 


y_ 


6 


7 


G 


9 


O \\ 712 






\ 










/ 






> 


\ 






V 









N 


\ 


/ 







through 12 successive positions of a slide valve, corresponding 
to intervals of 30 of the crank, beginning at the inner dead 
point. It is required to analyse the motion so as to express, 
in the form of a series of harmonics, the displacement of the 
valve from its mean position. (Practical Mathematics y p. 459.) 

Were the curve divided more symmetrically by the x 
axis, we should suspect comparatively little deviation from 
the first harmonic, i.e. an ordinary sine curve. But, fairly 
obviously, it is compounded with other harmonics as well. 

Run off the lengths of the ordinates to the edge of a 
paper strip as shown in fig. 263. Use the strip for plotting 
the points in fig. 264, but first reverse it, so that point 8 
is at the top and point 2 at the bottom. 



WAVE MOTION 



467 



For the first harmonic. Let of the strip coincide with O 
in fig. 264, and mark off these distances: to 6 on the ordinate 
through O, 1 to 7 on the ordinate through 1, 2 to 8 on the 
ordinate through 2, 3 to 9 on 3, 4 to 10 on 4, 5 to 11 on 5 (six 
measurements in all). Observe that these distances on the 




Fig. 264 

strip give, successively, y Q - y^ y 1 y 7 , y 2 j; 8 , y 3 y g , 
y ~~ Jio> Vb ~~" Vii- Draw a curve through the points, then 
the second half of the curve, below, through points obtained 
from the same measurements, reversed. Draw a tangent to 
this curve at a maximum or minimum point, HK, MN. 
The amplitude a' is half the distance from the tangent to 
the axis; it is 7/2 = 3-5. 



468 



CRAFTSMANSHIP IN MATHEMATICS 



The magnitude of the angle a can be obtained by measuring 
the length Op. The distance O to 6 = 180; hence Op = 151-8. 
Thus a = 180 - 151-8 - 28-2. 

For the second harmonic. The successive distances for 
the ordinates to be taken from the strip are (0 to 3) + (6 to 9), 
(1 to 4) + (7 to 10), (2 to 5) + (8 to 11). Observe that these 
distances are (y - y 3 ) + (y 6 - JVg), CVi Vi) + Cv? ~ J>io)> 
(yz ~~ y&) + (y& ~~ y\\)> Draw a curve through the points, 
and repeat below; and do the same thing again for the 
second period of the wave. 

To obtain the amplitude a", draw a tangent, and take 
J of the distance to the x axis, = -25. To obtain the angle 
, measure O?; Oq - of O3 = 150, hence = 210. 

For the third harmonic. The successive distances for the 
ordinates to be taken from the strip are, first (0 to 2) + (4 to 6) 
+(8 to 10), then (1 to 3) + (5 to 7) + (9 to 11). The curve 
almost coincides with the x axis, and as the distance to the 
crests has to be divided by 6 to obtain the amplitude a'", 



210* \ 

R--25J 




\ 



Fig. 265 

it is evident that this third term of the harmonic series is 
negligible. Hence the equation may be written: 

y = 3-5 sin(* + 28-2) + -25 sin(2^ + 210). 

Now draw the two harmonics to scale (fig. 265), recompose, 
compare the result with the original graph, and thus check 
the work. 



WAVE MOTION 



469 



Boys are always keen to know what is behind such un- 
usual procedure. The explanation is really very simple. 

Let (i), fig. 266, be the first harmonic (the fundamental), 
(ii) the second harmonic, (iii) the third, and (iv) the fourth. 
We can cut the whole wave in (ii) into two equal and similar 
parts, and slide the right-hand half along the axis and superpose 
it on the left-hand half. We may cut (iii) into three equal and 





Fig. 266 



similar parts, slide the second and third parts along and 
superpose them on the first. We may cut (iv) into four 
equal and similar parts, and again superpose. 

Now suppose we have a compound curve of unknown 
composition. If it consisted of the first harmonic only, it 
would be just a simple sine curve, like fig. (i). 

If the second harmonic is present, fig. (ii) represents that 
component. To test for its presence, cut the composite curve 
to 2?r into two, slide along and superpose, add the corre- 
sponding ordinates of the two parts thus superposed (yo+JVe* 
y l + y 7 , &c., algebraically, of course), take the average of 
each of these sums by dividing by 2, and plot the curve. 
That curve is the second harmonic together with any of its 
multiples, if any of these are components; but the curve 



470 CRAFTSMANSHIP IN MATHEMATICS 

does not contain any other harmonic than these multiples; 
i.e. the curve so obtained is, 

y = a 2 s'm(2x + oc 2 ) + 4 sin(4# -f a 4 ) -f &c. 

If the third harmonic is present, fig. (iii) represents the 
component. To test for its presence, cut the composite curve 
to 2?7 into three, slide along and superpose, add the corre- 
sponding ordinates of the 3 parts thus superposed, take the 
average of each sum by dividing by 3, and plot the curve. 
The curve is the third harmonic together with any of its 
multiples, if any of these are components, but the curve 
does not contain any other harmonic than those multiples; 
i.e. the curve obtained is, 

y = 3 sin(3# -f oc 3 ) + a 6 sin(6# -f a e ) -f &c. 

So with harmonics beyond the third. But these are 
rarely required; they affect the result too slightly. The 
proofs of these rules are very simple, and should be given. 

Inasmuch as there is no advantage in giving for analysis 
any composite curves containing harmonics beyond the 
third, this graphic work need not be carried further. But 
the boys ought now to return to the example represented by 
figs. 263 and 264, and penetrate the mystery of the paper 
strip: the additions from the strip are really the additions 
of superposed ordinates resulting from cutting up the com- 
posite curve, sliding to the left, and superposing. The 
reversal of the strip is readily seen to be a simple device for 
converting subtraction into addition. 

Teachers who think well of this method of Professor Runge 
may refer to Zeitschrift fur Mathematik und Physik, Vol. 48, 
443-56. 



Professor Nunn's Plan: the Principle 

A much more important curve- decomposition method 
may be briefly considered. The fundamental principle 



WAVE MOTION 



471 



underlying it is the obvious fact that the total area of either 
a complete sine curve or of a complete cosine curve is zero, 
since it is equally divided by the x axis. Professor Nunn's 
exposition (Algebra, pp. 521-3) is particularly illuminating, 
though I have sometimes found Sixth Form boys, who had 



Fig. 267 

not had a good training in solid geometry, puzzled over the 
geometrical figures. I append an outline of the exposition, 
together with a few new " solid " figures. 

On one side of a line AB of length /, draw the semi- 
sine curve y = a sin-*, 

choosing any value for 
the amplitude HK (= 
a). On the other side 
draw similarly the curve A 

A 

y = sin-*, with ampli- 
tude KL (= unity). Cut 
the figure out and fold 
it about AB until the 
planes of the two curves Fig. 2 68 

are at right angles. Now 

mould a solid, in clay, plasticine, soap, or any similar soft 
material, to fill up the space between the curves. 

In practice, the best way to do this is first to mould a 
rectangular prism / units long with cross-section KH X KL. 

Then draw the curve a sin ^* on the face of the prism 

FCDE (i), and the curve sin-* on the top of the prism 
MFEG (ii). Pare off horizontally round the curve as in (i), 




r- 



472 



CRAFTSMANSHIP IN MATHEMATICS 



and vertically round the curve as in (ii). The result is (iii), 
the solid we require, ALBH; the plan of the solid is the 



77 



figure bounded by AB and the curve sin-*, and the elevation 
in the figure bounded by AB and the curve a siny#. 




It is important to note that any section of the solid by a 
plane at right angles to AB is a rectangle (e.g. RSTV) whose 

adjacent sides are a sin-# and sin-#, x being the distance 

i if 

of the section from A. Note that the two lengths may be 
measured either on the flat surfaces behind and below or 
on the curved surfaces in front and above. Unless the solid 
is actually constructed, many boys will have difficulty in 
seeing this. 

The area of the section = a sin nx/l X sin nx/l 
= a sm 2 Ttx/l 

= 2(1 -cos2nx/l) 



i-:7* 



WAVE MOTION 473 

that is, the area of any section of the solid is equal to the 

algebraic difference between a constant area - and a variable 

2 
a 

area -cos 27rx/l. 
2 

For convenience, each of these areas may be looked upon 
as rectangles, each of height a/2. Thus the base of the 
former would be unity, and that of the latter 



I 



B 



(H) 





cos 2-ncc/l 

I ' 



Fig. 270 



The two rectangles may be regarded as cross-sections 
of two new solids of length AB ( /) and of uniform height 
a/2. Above are their plans. Note the neat, though obvious, 




(i) 



(ii) 



device for showing the width of the second. Fig. 271 shows 
perspective sketches (for the sake of clearness, figs. 270 and 
271 are drawn very considerably out of proportion, compared 
with figs. 268 and 269). 



474 CRAFTSMANSHIP IN MATHEMATICS 

Let a plane at right angles to AB cut the solids at PiP 2 > 
corresponding to RSTV in fig. 269 (iii). Then the section 
RSTV is equal to the difference between the sections P 1 and 
P 2 ; and so with any other vertical section. At KL in fig. 269 
(iii), the difference is between Q x and Q 2 , but since Q 2 is 
negative, the difference is the arithmetical sum. This is as 
might be expected, for the section on HL is the full section 
of the original rectangle. In the case of section P 2 , the width 
cos 27rx/l is positive; in the case of Q 2 , it is negative. Thus 
the area of the section P 2 must be reckoned positive and that 
of Q 2 negative. 

It follows that the part of the solid above AB in fig. 270 
(ii) must be reckoned positive, and that below AB negative. 
Hence we must regard the total volume of the solid in fig. 
271 (ii) as 0. But the volume of the solid in fig. 269 (iii) is 
equal to the difference of the volumes of the two solids in 
fig. 271 (i) and (ii). Hence the volume of the solid in fig. 269 
(iii) is equal to the volume of the simple prism in fig. 271 (i). 
The volume of the solid in fig. 269 (iii) is therefore al/2. This 
result is always a surprise to the boys, and they are much 
inclined to question it. They should be made to think about 
it carefully and to search for the fallacy they suspect. It 
will pay to make the boys work out one or two particular 
cases. Let them bear in mind that the volume of the rect- 
angular blocks in fig. 269 (see fig. 268) is / X a X 1 = al. 

On one or two occasions I have known Sixth Form 
boys cut out their models so carefully that, when checked 
by weighing, the results have been surprisingly accurate. 
To cut fig. 269 (iii) out of soap, and to weigh the model 
against the parings, may afford a very convincing check. 



WAVE MOTION 



475 



The Principle Applied 

Consider the following figure, one complete element 
(O to 2?r) of a composite wave. The problem is to determine 
the amplitudes of the various component harmonics; that 
done, the harmonics are easily drawn. Since the right-hand 
half of the curve is the " image " of the left-hand half, it is 
sufficient to consider the left-hand half alone; call its length 
/. We will assume that there are two components, viz. y = a v 
simrx/l and y = a 2 smS^x/l, in other words that the given 
curve is made up of the first and second harmonics. (We 
know from the kind of symmetry that the third harmonic is 
not a component (see p. 463).) 

On the line IVTN' (= MN = /), draw the curve y = 
sin irx/l inverted, i.e. a sine curve with amplitude unity; and 




Fig. 272 



make a model of the solid determined by the two curves 
when the lines M'N' and MN are made to coincide and 
the planes of the figures are at right angles. Note that any 
section FGK at right angles to MN is rectangular, as in the 
solid of fig. 269. The solid is not an easy one to model 
accurately (fig. 273). 

The volume of the composite solid is equal to the 



476 CRAFTSMANSHIP IN MATHEMATICS 

sum of the two solids determined by the curves, 

(i) y = smnx/l and y = a^ smnx/l, 
and (ii) y = sin roe// and y = a z s'm2nx/l. 

But the latter of these volumes is easily proved equal to 
(cf. fig. 271 (ii)), and the volume of the former is a x //2 (cf. 
fig. 271 (i)). Hence 

total volume of solid = aJ/2 (i) 




Fig. 273 

But the volume may also be determined directly, by 
calculating the mean value of its cross-section. Consider, for 
instance, the vertical section at G on MN where MG = 2//3 
= x, so that TTX/I = 277/3 radians or 120. The section in a 
rectangle whose sides FG, GK are closely analogous to the 
sides RS, ST in fig. 269 (iii). Of these two sides, FG may 
easily be determined by actual measurement from the curve, 
while GK = sin!20 = \/ 3 / 2 - The product gives the area 
of the vertical section through FGK. 

In this way we may find the area of any number of such 
vertical sections. For convenience, divide MN into 12 equal 
parts, Calculate the areas of the respective sections through 



WAVE MOTION 477 

the dividing points, and then by Simpson's rule* the volume 
of the solid. Deduce from this the average cross -section 
A x by dividing by /. 

Thus vol. = A!/. 
But vol. = aJ/2 (by (i)). 

" ll - A / 
T~ Al/ * 

or flj = 2Aj. 

In a similar manner, by supposing a second solid to be 
formed by combining the given half curve with the curve 
y = sin2?rjc//, the value of a 2 may be determined. If the 
given curve contained any other harmonic components, 
their amplitudes might be determined in the same way. 

The principle of the method is that any sine curve y = 
sin rnx/l when combined with half the given composite curve 
determines a solid whose volume (a r l/2) depends on the 
amplitude a r of the component y = a r sin mx/l, and not at 
all on the amplitude of any other component. In this way, 
the successive sine components can be dealt with one by 
one, and their amplitudes determined. The determination 
of the amplitudes is, of course, the very essence of the problem. 

The work of computing the average cross-sections can be 
divided up amongst the members of the class. Instruct them 
to carry out the following operations, and to tabulate the 
results: 

(1) To determine the amplitude of the first harmonic. 

(a) Divide up MN into twelve 15-phase differences; 
erect the ordinates and measure their lengths in millimetres. 
In accordance with Simpson's rule, only half the height 
of the first and last ordinates is required in the calculations, 
but as, in this instance, these happen to be zero, the halving 
makes no difference. 

(/J) Calculate the successive values of sinn!5, n being the 
number of the ordinates. 

* " Add half the first and last areas and the whole of the intermediate areas, 
and multiply the sum by the common interval." 



478 



CRAFTSMANSHIP IN MATHEMATICS 



(y) Multiply (a) by (/?) and so obtain the areas of the 
successive sections of the solid (fig. 273). 



i. 


ii. 


1st Harmonic. 


2nd Harmonic. 


Number 
of 
Ordinate. 


Length of 
Ordinate 
in mm. 


iii. 


iv. 


V. 


vi. 


vii. 


viii. 


Angle. 


Sine. 


Area 
ii or iv. 


Angle. 


Sine. 


Area 
ii or iv. 























+o 


1 


13-0 


15 


26 


3-38 


30 


5 


+ 6-50 


2 


23-0 


30 


5 


11-50 


60 


87 


+ 19-93 


3 


29-5 


45 


71 


20-90 


90 


1-0 


4-29-50 


4 


31-5 


60 


87 


27-40 


120 


87 


4-27-40 


5 


29-5 


75 


97 


28-60 


150 


5 


4-14-75 


6 


24-0 


90 


1-0 


24-00 


180 








7 


17-5 


105 


97 


16-90 


210 


5 


-8-75 


8 


10-3 


120 


87 


8-92 


240 


-87 


-8-92 


9 


5-0 


135 


71 


3-54 


270 


-1-0 


-5-00 


10 


2-0 


150 


5 


1-00 


300 


-87 


-1-74 


11 


1-0 


165 


26 


26 


330 


-5 


-50 


12 





180 








360 








Total = 
Average area A x = 


146-40 


Total = 


73-16 


12-20 


Av'ge = A 2 = 


6-097 



Amplitude of the 1st harmonic = 



fll = 2Aj = (12-2 mm. X 2) 
= 24-4 mm. 
= 2-44 cm. 



(2) To determine the amplitude of the second harmonic. 
Corresponding to the half curve MN will be a complete 
sine curve of the second harmonic. Hence the angles will 
now be #30, and the sines from 180 to 360 will be negative 
The ordinate lengths will be the same as before. 



Amplitude = a a = 



= (6-097 mm. X 2) 
= 1-22 cm. 



WAVE MOTION 479 

Hence the original curve is, 

y = 2-44 sinnx/l+ 1-22 sin2nx/l. 

The periods being known, and the amplitudes having been 
found, the angles follow at once. 

Let the boys realize fully that the essence of the problem 
is the discovery of the amplitudes of the component harmonics. 



31-5 




274 



By improvising the solids and devising two different schemes 
for determining their volumes, we obtain two different formulae 
each involving a in terms of A. It is true that A appears as 
an area, but, by taking one of the dimensions of the area as 
unity, A becomes a linear value, and of course we begin by 
giving the solid a base consisting of a sine curve of unit 
amplitude. 

The subject can be followed up by integration. The boys 
are now ready for it, for they have learnt what the subject 
is really about. 

Books to consult: 

1. The Teaching of Algebra, Nunn. 

2. Manual of Practical Mathematics, Castle. 

3. Any modern standard work on Sound. 



480 CRAFTSMANSHIP IN MATHEMATICS, 

CHAPTER XXXII 

Mechanics 

The Teacher of Mechanics 

The most successful teachers of mechanics whom I have 
known are those who have had a serious training in a me- 
chanical laboratory; who know something of engineering, and 
are familiar with modern mechanism; who are competent 
mathematicians; and who have mastered Mach's Mechanics, 
especially Chapters I and II.* Mach's book is universally 
recognized as the book for all teachers of mechanics. It 
deals with the development of the fundamental principles 
of the subject, traces them to their origin, and deals with 
them historically and critically. The treatment is masterly. 
The book might with advantage be supplemented by Stallo's 
Concepts of Modern Physics (now out of date from some 
points of view), Karl Pearson's Grammar of Science, and 
Clifford's Common Sense of the Exact Sciences and Lectures 
and Essays (still first-rate, though written 50 years ago). 

It is of great advantage to a teacher of mechanics to be 
familiar with the subject historically. The main ideas of the 
subject have almost always emerged from the investigation of 
very simple mechanical processes, and an analysis of the 
history of the discussions concerning these is the most 
effective method of getting down to bedrock. 

Who were the great investigators? The scientific treat- 
ment of statics was initiated by Archimedes (287-212 B.C.), 
who is truly the father of that branch of mechanics. The 
work he did was amazing, but there was then a halt for 1700 
or 1800 years, when we come to Leonardo, Galileo, Stevinus, 
and Huygens; to Torricelli and Pascal; and to Guericke 
and Boyle. For dynamics, we go first to its founder Galileo 

* Hertz also wrote a Mechanics of the same masterly kind, but there is no English 
translation, so far as I know. 



MECHANICS 481 

(falling bodies, and motion of projectiles), then to Huygens 
(the pendulum, centripetal acceleration, magnitude of acceler- 
ation due to gravity), and then to Newton (gravitation, laws 
of motion). The great principles established by Newton 
have been universally accepted almost down to the present 
time, and, so far as ordinary school work is concerned, will 
continue to be used at least during the present generation. 

A boy is always impressed by Newton's argument that 
since the attraction of gravity is observed to prevail not 
only on the surface of the earth but also on high mountains 
and in deep mines, the question naturally arises whether it 
must not also operate at greater heights and depths, whether 
even the moon must not be subject to it. And the boy is 
still more impressed by the story of the success of Newton's 
subsequent investigation. 

Newton's four rules for the conduct of scientific investi- 
gation (regulce philosophandi) are the key to the whole of his 
work, and should be borne in mind by his readers. 

The First Stage in the Teaching of Mechanics 

How do successful teachers begin mechanics with boys of 
about 12 or 13? They usually begin by drawing upon the 
boys' stock of knowledge of mechanism.* Most boys know 
something of mechanism, some will have had enough curiosity 
to discover a great deal, and a few will probably have had 
experience of taking to pieces machines of some sort and of 
putting them together again. This stock of knowledge may be 
sorted out, and the topics classified and made the subjects 
of a series of lessons. By means of an informal lesson on some 
piece of mechanism, an important principle may often be 
worked out, at least in a rough way. 

I have known a teacher give his first lesson on mechanics 
in the school workshop, utilizing the power-driven lathe and 
the drilling-machine; another first lesson in the school play- 
ground, an ordinary bicycle being taken to pieces. I have 

See Chapter VIII, Science Teaching. 
(291) 32 



482 CRAFTSMANSHIP IN MATHEMATICS 

seen a model steam-engine used for the same purpose, 
and I have known beginners taken to a local farm to watch 
agricultural machinery at work. In all these instances the 
boys learnt that their new subject seemed to have a very 
close relation with practical life. They were not made to 
look upon it as another branch of mathematics, and a rather 
difficult branch at that. 

Let the early lessons be lessons to establish very simple 
principles. Never mind refinements and very accurate 
measurements. Do not bother about small details, and avoid 
all complications. Let the boy get the idea, and get it clearly. 
Very simple arithmetical verifications are quite enough at 
this stage. The boy's curiosity is at first qualitative; let 
that be whetted first, and then turned into a quantitative 
direction gradually. Encourage the boy to find out things for 
himself, and do not tell him more than is really necessary. 
Encourage him to ask questions, but as often as possible 
answer these by asking other questions which will put him 
on a new line of inquiry. Let him accumulate knowledge 
of machines and machine processes. Give him some scales 
and weights, and a steelyard, and tell him just enough to 
enable him to discover the principle of moments, but do 
not talk at first about either " principle " or " moments ". 
It is good enough if at this stage he suggests that 

long arm X little weight - short arm X big weight. 

He has the idea, and the idea is expressed in such a form that 
it sticks. Give him a model wheel and axle, give him a hint 
that it is really the lever and the lever-law over again, and 
make him show this clearly. Give him some pulleys and let 
him discover, with the help of one or two leading questions, 
how a small weight may be made to pull up a big weight, 
and let him work out the same law once more, but now in the 
form that what is gained in power is lost in speed. Give him 
a triangular block and an endless chain, let him repeat Ste- 
vinus' experiment, and so discover the secret of the inclined 
plane. Let him use a jack to raise your motor-car (and inci- 



MECHANICS 483 

dentally learn something about " work "); now tell him some- 
thing about the pitch of the screw, something about Whit- 
worth's device for measuring very small increases in length, 
something about the manufacture of a Rowlands grating. 
Encourage him to give explanations of mechanical happenings 
in everyday life, and use his suggestions as pegs on which to 
hang something new. 

A term of this kind of work pays. The boy is accumulating 
knowledge of the right sort, and when the subject is taken up 
more formally and with a more logical sequence, rapid progress 
may be made. Once he has been taught to read elementary 
mechanism, it is easy enough to teach him its grammar. 
Surely this is the right sequence. Mechanism must come 
before mechanics. The mathematics of the subject is a super- 
structure, to be built upon a foundation of clear ideas. 

Of course, if the preliminary work of the preparatory school 
or department has been properly done, the way is paved for 
an earlier treatment of a more formal kind. 

The Second Stage 

The second stage should consist of work of a more syste- 
matic character, but still work essentially practical, though 
arranged on a logical string. Ideas will now be classified, and 
mathematical relations gradually introduced. But the physical 
thing and the physical action must still remain in the front of 
the boy's mind. The mathematics will take care of itself. 

Let the teaching be inductive as far as possible. Obtain 
all .necessary facts from experiments, and do not use experi- 
ments merely for verifying a principle enunciated dogmatically. 

The basic principles to be taught are really very few, and 
a boy who knows these thoroughly well can work most ordinary 
problems on them. Mechanics is, after all, largely a matter of 
common sense. The laws of equilibrium, together with the 
ratio of stress to strain, covers almost the whole range of 
statical problems, including those of hydrostatics; while New- 
ton's Laws of Motion covers practically everything else. But 



484 CRAFTSMANSHIP IN MATHEMATICS 

of course these are basic principles. If they are known, 
known, derived principles are learnt easily enough; if they are 
only vaguely known, derived principles are never really 
mastered. 

Statics or dynamics * first? Teachers do not agree. There 
is much to be said for beginning with dynamics, first using 
the ballistic balance for studying colliding bodies, and the 
momentum lost by one and gained by another; it is then an 
easy step to pass on to the idea of force. But a boy who is 
led to think of a force as something analogous to muscular 
effort will always be in trouble, and in any case he is likely 
to form a very vague idea of acceleration. Of course, uniform 
acceleration is anything but common in practical life: we 
nearly always refer either to falling bodies or to a train 
moving out from a station. And it is this difficulty that 
makes many teachers take up statics first. Although, at the 
outset, a boy's working idea of force is necessarily crude, 
a spring balance, for simple quantitative experiments, helps 
to put the boy on the right track, and there is much to be 
said for allowing him to assume, to begin with, that weight 
is the fundamental thing to be associated with force. At 
an early stage he may verify, to his own satisfaction, the 
principles of the parallelogram and triangle of forces, but he 
must be warned that he has not yet " proved " these principles 
and cannot yet do so. But since the parallelogram of forces is 
such a useful working principle, it would be foolish not to 
allow the boy to use it before he can prove it formally. At 
this stage formal proofs are difficult, and it is simply dis- 
honest to encourage a boy to reproduce a page of bookwork 
giving a proof of something quite beyond his comprehension, 
though this was common enough thirty or forty years ago. 

Do not employ graphic statics at too early a stage, or the 
real point at issue may be obscured. 

Now as to dynamics. What is the best approach? We 
have already referred to the ballistic balance. Should At- 
wood's machine be used? It may be used, perhaps, for 

* The terms kinetics and kinematics are falling into disuse. 



MECHANICS 485 

illustrating the laws of motion, but not as a practical method 
of finding g. 

Atwood's machine has been superseded by Mr. Fletcher's 
trolley,* by means of which practically the whole of the prin- 
ciples of dynamics may be satisfactorily demonstrated. It 
lends itself to many experiments, all of which provide a 
space-time curve ready made, and, from that, speed-time 
and acceleration-time curves may be plotted. In a paper 
read at the York meeting of the British Association, Mr. C. E. 
Ashford gave details of a large number of trolley experiments 
as performed at Dartmouth, a school where the teaching 
of mechanics is well known to be of a high order. Reference 
should be made to Mr. Fletcher's own article in the School 
World for May, 1904. In it he shows how boys may be 
given sound ideas of the physical meaning of the terms, 
moment of inertia, angular momentum, moment of momen- 
tum, and therefore of moment of rate of change of momentum 
and moment of force. Useful teaching hints may also be found 
in Mr. S. H. Wells's Practical Mechanics and Mr. W. D. 
Eggar's Mechanics. 

Once the foundations of mechanics have been well and 
truly laid the superstructure may be erected according to 
traditional methods. To leave the subject just as developed 
in the laboratory would be to leave it unfinished. But the 
superstructure may now be built properly. When necessary 
formulae have been evolved from experiment, the physical 
things behind the formulae have to the boy a reality of mean- 
ing which the older " methods of applied mathematics " 
teaching could not possibly give him. 

If principles are not understood, proofs have no meaning. 

Throughout the whole of a mechanics course every oppor- 
tunity should be taken to excite the boys' interest in new 

mechanical inventions. It helps the more academic work 



*The friction of the trolley may be eliminated either by tilting the plane to the 
necessary angle, or by attaching a weight that will just maintain uniform motion. 
The friction of the pulley over which the thread passes cannot be compensated, and 
it is therefore necessary to use a good pulley. 



486 CRAFTSMANSHIP IN MATHEMATICS 

enormously, and makes the boys feel that the subject is really 
worth taking trouble over. Examples occur on every side 
variable speed gears, transmission gears, taximeters, boat- 
lowering gear, automatic railway signalling, automatic tele- 
phones, the self-starter in a motor-car, the kick-starter in a 
motor-cycle, and so on. Some mechanical devices depend, in 
their turn, on electricity, and their place of introduction into 
a teaching course would be determined accordingly. Complex 
mechanisms like the air-plane, the submarine, the paravane, 
should not be wholly forgotten. Boys can read up such things 
for themselves, and perhaps prepare and read papers on them 
to the school science society. 

Hydrostatics 

The mechanics of fluids is an exceedingly difficult subject 
to teach effectively. Even a Sixth Form boy is sometimes held 
up by questions on the barometer or on Dulong and Petit's 
equilibrating columns. The work of Archimedes and Pascal 
for liquids and of Boyle for gases cannot be too well done. 
Above all, the U-tube must receive careful attention, and 
especially the surface level above which pressures are compared. 
Do not buy Hare's apparatus from an instrument-maker's. 
The standard pattern is always made with two straight tubes, 
of the same bore, fixed vertically. Let the boys make a variety 
of forms of this apparatus for themselves, and work out the 
vertical height law from data as varied as possible. Approach 
the whole subject of hydrostatics from the point of view of 
familiar phenomena, e.g. measure the water pressure from a 
tap in the basement and again from a tap in the top story of 
the school, and see if there is any sort of relation between the 
difference of these pressures and the height of the school. 
Do not try to establish a principle formally until the phenomenon 
under investigation is clearly understood as a physical happening. 
Let boys know really what they are going to measure before 
they begin to measure.* 

The preceding paragraphs are taken from Science Teaching (pp. 121-8). 



MECHANICS 487 

The Johannesburg British Association Meeting 

At the Johannesburg meeting of the British Association, 
an animated discussion took place on the general question 
of the teaching of mechanics. It followed on a paper read by 
Professor Perry. We append a few suggestive extracts. 

Professor Perry. " The very mathematical man often 
does not know anything of mechanics; it is the subject of 
applied mathematics that he has studied and that he cares 
for. 

" The two elementary principles of statics, (1) if forces 
are in equilibrium, their vector sum is zero, and (2) the sum 
of their moments about any axis whatsoever is zero, ought 
to be so clear to a pupil that it is practically impossible for 
him to forget them. They ought to be as much a part of his 
mental machinery as the power to walk is part of his physical 
function. 

" I lay no stress upon mere abstract proofs of propositions 
in mechanics. When understanding is affected there is no 
difficulty about the proofs. It is quite usual to find men who 
can prove everything, without having any comprehension of 
what they have proved/' 

Mr. W. H. Macaulay. " I agree with the taking of statics 
before dynamics. I also agree that graphical statics is a subject 
full of dodges, though very good to learn if you want to use 
them every day." 

Professor Boys. " I absolutely agree as to the desirability 
of dealing with fundamental principles, and of not worrying 
about innumerable details. ... A friend of mine heard 
Lord Kelvin say in one of his lectures, ' And now we come 
to the principle of the lever. You will understand that levers 
are divided into three orders, levers of the first order, of the 
second, and of the third but which of them is which I 
cannot for the life of me tell you/ Textbooks were at one 
time filled up with futile and unnecessary kinds of dis- 
crimination which had nothing whatever to do with the sub- 
ject." 



488 CRAFTSMANSHIP IN MATHEMATICS 

Professor Bryan. " The idea of mechanics which appeals 
most readily to a young boy is that it has something to do 
with machines, and that machines have something to do with 
turning out useful work. There is no better way of stimulating 
interest in the subject than showing the beginner that when 
you have got your machine for changing one kind of work 
into another, you are no better off than when you started. " 

Professor Hicks. " My own experience is in approaching 
mechanics from a kinetic point of view. First let the boys 
find out by experiment that momentum remains constant. 
Of course the first thing depends on what mass is; then we 
must proceed to show that when two bodies collide with equal 
velocities they come to rest. By making experiments of 
velocities of colliding bodies, boys get to realize that momentum 
remains unalterable. Given two colliding bodies in a straight 
line, the momentum lost by one is gained by the other. 
By getting a large number of experiments, pupils come to 
a realized knowledge of that." 

Sir David Gill. " I remember Clerk Maxwell illustrating 
the misuse of definitions by a funny story. He said he went 
into his room one day, and there was a white cat which 
jumped out of the window. He and his friends ran to the 
window to see what had become of the cat, and the animal 
had disappeared, no one being able to solve the mystery. 
At last he solved the problem. He said it must be this. The 
white cat jumped out of the window, fell a certain distance 
with a certain velocity, and collided with an ascending black 
cat. There were therefore two equal and opposite cats 
meeting with equal and opposite velocities, the result being 
no cat. Without a proper understanding of definitions of 
these things, one might arrive at such an absurdity as this 
story illustrates." 

Professor Forsyth. " The first stage in teaching mechanics 
is not the stage in which pupils have to prove, or attempt to 
prove, or can be expected to prove, anything. That belongs 
to a later stage. The first thing to do is accustom the pupils 
to the ordinary relations of bodies and of their properties." 



MECHANICS 489 

Mr. W. D. Eggar. " I should like to see a penny-in- 
t he-slot automatic weighing machine in every passenger lift, 
so that the fundamental experiment of showing a connexion 
between force and acceleration could be within the reach of 
everybody." 

Professor Minchin. " I hope to see the term * centrifugal 
force ' utterly banished." 

Mr. C. Godfrey. " Statics is a fairly easy matter if one 
begins with experiment. Nor need experiment cease after 
the first stage; any school should be able to get hold of some 
bit of machinery with plenty of friction in it, say a screw- 
jack, and investigate efficiency. Plotting ' load ' against 
* effort ' leads to very striking results. 

" There is the question of mass and weight. In vain one 
resorts to the centre of the earth; it is all too hypothetical. 
I remember as a boy being puzzled to understand how the 
weight of a train (acting vertically) could have anything to 
do with its acceleration under a pull (horizontal) from the 
engine. 

" We might give a touch of reality to the kinetics course 
by brake horse-power determinations. It should be possible 
to rig up for a few shillings a brake-drum on a motor (electric 
or water); even a motor-cycle on a stand or a foot-lathe 
might serve the purpose. 

" Engineers talk in a very confusing way about centrifugal 
force. When a particle moves in a circle uniformly, the force 
on the particle is centripetal and the force on the constraints 
is centrifugal. But the popular use of language and the 
popular belief is that there is an outward force on the particle." 

" Applied " Mathematics 

The old school of " pure " mathematicians very cleverly 
picked out from the whole subject of mechanics and engineer- 
ing such problems as lent themselves to algebraic and 
geometrical treatment, and left the residue, rather disdain- 
fully labelled " applied mechanics ", to be dealt with by 



490 CRAFTSMANSHIP IN MATHEMATICS 

teachers of lower degree. Note the term " applied ". The 
real mechanics was the mechanics that could be done from 
an easy chair, and was a mathematicians' job. The building 
of the Assouan dam and of the Forth Bridge were trivial 
things which any " ordinary engineer " could take in hand, 
trivial things that had no relation whatever to " pure " 
thought. This temper survived even until the present century. 
When the two Wrights were risking their lives by experiment- 
ing with the first air-plane, a well-known mathematician wrote 
to the press protesting against such folly, inasmuch as mathe- 
maticians had not yet worked out the mathematical principles 
of flight! 

The mathematician's proper share of such work is to 
begin where the inventor or the engineer leaves off; it is 
not his business to invent paper air-planes, but to learn 
from the real thing the principles of flight and to see if these 
rest on secure mathematical foundations; if they do not, 
he may be able to offer fruitful suggestions. Of course if 
the mathematician happens to have been trained as an 
engineer, that is a different matter. Unless the teacher of 
mechanics knows something of actual engineering, his me- 
chanics is likely to have but a remote connexion with actual 
mechanism. There are still teachers of mechanics who have 
had neither workshop nor laboratory experience, and naturally 
they tend to shirk those parts of the subject that do not 
come within the four corners of algebra and geometry. It 
is not an uncommon thing for a course of lessons on ele- 
mentary statics to include not a single word about, for instance, 
the equilibrium and stability of walls, the effect of buttresses, 
the thrust along rafters, or about roof-trusses or cranes. 
Friction may be the subject of a lesson with no mention 
whatever of lubricants. Energy may be the subject of others, 
and yet no reference be made to energy storage in, for example, 
accumulators and fly-wheels. The transmission of motion 
and power is rarely touched upon seriously in a course of 
mechanics lessons. And yet all such things as are thus ignored 
are just those things that have already been included, in 



MECHANICS 491 

some measure, within the four corners of the boys' daily 
experience. Subjects like tension and compression, shearing 
and torsion, beams, girders, and frameworks, are passed 
over hurriedly as of little importance. Why is elementary 
hydrostatics so often given such short shrift? Why is it not 
followed up by the subject which really matters, viz. ele- 
mentary hydraulics the flow of water through orifices and 
pipes, the pressure in a water-main, water-wheels, turbines, 
the propulsion of ships and air-planes, and hydraulit machines? 
As for capillarity and surface tension, which lend themselves 
to all sorts of delightful experiments, they are too often an 
affair of just blackboard and chalk. Do not put off that 
interesting section of physics, " properties of matter " (the 
twin-sister of mechanics), until the Sixth Form. The mathe- 
matics of it in the Sixth, yes; but the necessafy laboratory 
course can be taken in the Fourth and Fifth. 

In short, the mathematics of mechanics is very serious 
Sixth Form work. The practical work that must be done 
before the mathematical work can profitably be attempted 
may be done earlier. 

We will give an extract from an elementary textbook 
on Mechanics for Beginners, with a very well known name 
on the title-page. It is an introduction to Moment of 
Inertia. 

" Let the mass of every particle of a body be multiplied 
into the square of its distance from an assigned straight line; 
the sum of these products is called the moment of inertia of 
the body about that straight line. The straight line is often 
called an axis. 

" The moment of inertia of any body about an assigned 
axis is equal to the moment of inertia of the body about a 
parallel axis through the centre of gravity of the body, increased 
by the product of the mass of the body into the square of the 
distance between the axes. Let m be the mass of one particle 
of the body; let this particle be at A. Suppose a plane through 
A, at right angles to the assigned axis, to meet the axis at 




492 CRAFTSMANSHIP IN MATHEMATICS 

O, and to meet the parallel axis through the centre of gravity 
A at G. From A draw a straight line 
AM, perpendicular to OG or to OG 
produced. Let GM = x, where x is 
a positive or negative quantity accord- 
ing as M is to the right or left of G. 
"M By Euclid II, 12, 13, we have OA 2 
= OG 2 + Ga 2 + 2OG.*; therefore, 

m . OA 2 = m . OG 2 + m . GA 2 + 2OG . m . x. 

A similar result holds good with respect to every particle of 
the body. Hence we see that the moment of inertia with 
respect to the assigned axis is composed of three parts, 
namely, first the sum of such terms as w.OG 2 , and this will 
be equal to the product of the mass of the body into OG 2 ; 
secondly, the sum of such terms as m.Ga 2 , and this will be 
the moment of inertia of the body about the axis through 
G; and thirdly the sum of such terms as 2OG .m.x y which is 
zero. Hence the moment of inertia about the assigned axis 
has the value stated in the proposition. " 

Be it remembered that this book is a book for "beginners". 
I remember a Fourth Form once being given ten minutes 
to read up the subject-matter just quoted. Then came 
questions. Said one boy, " I thought inertia meant lazi- 
ness. " " So it does, a sort of laziness."" Then does 
' moment of inertia ' mean a moment of laziness?" Said 
another boy, " How are we to find the mass of one particle? 
Do we crush the thing up in a mortar, and weigh one of the 
particles? or do we weigh the thing first, then crush it up, 
count up the particles, and divide the weight by the number?" 
The teacher replied, " Don't be silly; moment of inertia is 
not real; it is only theory "! 

Who could blame the boys for asking such questions? 
How could they have obtained the faintest insight into the 
nature of the subject under discussion? 

Forty or fifty years ago, Todhunter's Analytical Statics 
was a standard work, used by mathematical students at the 



MECHANICS 493 

University. There are cases on record of men who obtained 
Firsts in mathematics but who in the subject mentioned had 
read no other book at all, had never handled a piece of appa- 
ratus in their lives. Fortunately that age has passed away. 

Mr. Fletcher's trolley, which is now in general use for 
teaching dynamics, is not always made so serviceable as it 
might be. (Readers should refer again to Mr. Fletcher's 
own comprehensive article in the School World for 1904.) 
In Perry's Teaching of Elementary Mechanics, already referred 
to, Mr. Ashford, formerly Head of the Royal Naval College, 
Dartmouth, gives some exceedingly useful hints on the 
further use of the trolley. 

The early teaching of mechanics must be given an ex- 
perimental basis. Mathematicians unacquainted with the 
mechanical laboratory should let the subject alone. It is 
better not taught at all than to be taught as mere algebra and 
geometry. Only if basic principles are established experi- 
mentally can the subsequent mathematical work be given 
a reality and a rigour that command respect. 

44 The Teaching of Mechanics in Schools " 

A report on " The Teaching of Mechanics in Schools ", 
specially prepared for the Mathematical Association, was 
issued in 1930. The responsible sub-committee was appointed 
in 1927 by the General Teaching Committee of the Association. 
The sub-committee included such well-known teachers as 
Mr. C. O. Tuckey, Mr. W. C. Fletcher, Mr. W. J. Dobbs, 
Mr. C. J. A. Trimble, and Mr. A. Robson, and the Report 
will therefore carry great weight amongst all teachers of 
mathematics. Every page reveals the hand of the practical 
teacher. No teacher of mathematics should fail to give it 
his serious attention. We quote a few short paragraphs in 
order that the reader may gather some notion of the general 
tenor of the Report. 

" There is perhaps no branch of mathematical instruction 
for which a pupil comes prepared with a larger body of 



494 CRAFTSMANSHIP IN MATHEMATICS 

intuitional knowledge than he does for mechanics. The 
suggestions made in this report are based on the view that 
this body of knowledge should form the foundation of the 
teaching, and that the aim of the teaching should be largely 
concerned with a development of a taste for such accurate 
thought and consideration of mechanical facts as will make 
them more intelligible, increasing the interest which attaches to 
the mechanical behaviour of things, and leading to that insight 
which brings this behaviour more completely under control/' 

" Just as geometry has its roots in familiar phenomena of 
daily life, so has mechanics. The basic principles of both 
sciences can be gathered, at least crudely, from ordinary 
observation this is the process we knew as abstraction. " 

" When we have carried the process some little way it 
becomes necessary, or at least economical, to arrange things 
so as to provide a more exact answer to a definite question 
than can be obtained from observation of unarranged or 
uncontrolled phenomena. So we get two processes, fading 
into one another no doubt in marginal cases, but in 
general easily distinguishable, viz. reflection on ordinary 
experience, and deliberately arranged experiment. In the 
former it may be noted and it is perhaps an essential part 
of the distinction experience comes before thought; we 
may or may not observe and reflect upon it and we may or 
may not make scientific use of it. In the latter, viz. experi- 
ment, as it has to be deliberately arranged, thought comes 
first we must frame a question before we can arrange the 
experiment which is to give the answer. " 

" While there is room for difference of opinion and 
practice as to the place of experiment in the school treatment 
of mechanics, there is no question that observation and 
reflection on ordinary experience are essential for any proper 
grasp of the subject. The widespread neglect of this obvious 
truth is responsible for much lack of success in the teaching 
of the subject." 

" In mechanics the crude facts lie open to direct observa- 
tion, and the role of experiment is limited to rendering more 



MECHANICS 495 

precise an answer which, in the rough, can be given without 
experiment." 

" The function of experience is to provide a basis of reality 
for the abstract science of the textbook and the schoolmaster, 
and the paramount duty of the latter is to make his pupils 
conscious of their own experience, to get them to reflect 
upon it, to co-ordinate their existing store and to open their 
eyes to observe more closely and to see the significance and 
interest of much that the unobservant mind ignores. Training 
of this sort is essential if the subject is to have its real 
value. ... In each fresh section of the work, the first thing 
to do is to collect and clear up existing experience bearing 
on the matter in hand." 

The " Contents " of the Report are as follows: 

1. Position in the Curriculum. 

2. General Aims. 

3. Experience and Experiment. 

4. Order of Treatment. 

5. The beginning of Statics. 

6. The beginning of Dynamics. 
7 Miscellaneous Topics: 

(i) Earlier teaching of Mechanics; (ii) Experiments; (iii) 
Initial difficulty of Statics; (iv) Kinematics; (v) Units and 
Dimensions; (vt) Horse Power; (vii) Formation of the 
Equations of Motion; (viii) Jointed Frames; (ix) Friction; 
(x) Torque, Couples; (xi) Geometrical and Algebraic 
Methods; (xii) Impact and the Lew of Momentum 
and Energy; (xiii) Rotatory Motion; (xiv) Limitations 
of School Dynamics; (xv) Miscellaneous. 

8. To examiners. 

9. Appendices: 

(i) Wheeled Vehicles; (ii) Momentum Diagram. 

Newtonian Mechanics superseded 

It is commonly said that Einstein has dethroned Newton, 
and this in a sense is true, inasmuch as Newton's laws 
have been superseded; but Einstein has always regarded 
Newton as his master. Improved instruments have led to 



496 CRAFTSMANSHIP IN MATHEMATICS 

the discovery of facts unknown to Newton, and Newton's 
laws have had to be amended in order that the new facts 
may be included, and this has been really Einstein's work. 

At the end of last century, physical science recognized 
three indisputable universal laws: (1) conservation of matter; 
(2) conservation of mass; (3) conservation of energy; and 
on the strength of these laws physical science became almost 
aggressively dogmatic. They should, of course, have been 
regarded merely as working hypotheses. Since 1905, it has 
been recognized that energy of every conceivable kind has 
mass of its own. Mass is the aggregate of rest-mass and 
energy-mass. Mass is seen to be conserved only because 
matter and energy are conserved separately. 

Then, again, as to the question of fixed axes. The trouble 
that some of us had when learning mechanics in the days 
of our youth arose (as we now see) from the assumption that 
axes were fixed in space. It is impossible not to feel that 
such able men as Kelvin, Tait, and Routh were not suspicious 
that the theory was in some way incomplete, but they seem 
to have acquiesced in giving to Newton's laws of motion 
a universality and finality which we now know the laws 
did not really possess. 

Listen to Clerk Maxwell (as a mathematician probably 
second only to Newton), in his lighter moments: 

"RIGID BODY (Sings) 

" Gin a body meet a body 

Flyin' through the air, 
Gin a body hit a body 

Will it fly? and where? 
Ilka impact has its measure, 

Ne'er a ane hae I, 
Yet a* the lads they measure me 

Or, at least, they try. 

" Gin a body meet a body 

Altogether free, 
How they travel afterwards 
We do not always see. 



MECHANICS 497 

Ilka problem has its method 

By analytics high; 
For me, I ken na ane o* them, 

But what the waur am I?" 

How are the tremendously far-reaching twentieth century 
changes to affect our teaching? Probably not at all except 
in the Sixth Form, for another twenty years to come. Of 
course the changes are very slight, too slight to affect appreci- 
ably the actual practice of mechanics. But the theory of 
mechanics is another story altogether. 

Books to consult: 

1. Mechanics, J. Cox. 

2. Introduction to the Principles of Mechanics, J. F. S. Ross. 

3. Theoretical Mechanics, J. H. Jeans. 

4. Mechanics of Fluids, E. H. Barton. 

5. Treatise on Hydrostatics, G. W. Minchin. 

Routh, and Lamb, should still be on every teacher's shelf. Elemen- 
tary books like Ashford, Eggar, and Fawdry, are full of useful teaching 
hints. The book for every teacher to master is Science of Mechanics 
(Mach). 



CHAPTER XXXIII 

Astronomy* 

Mathematics or Physics? 

If astronomy is included in the school physics course, 
the necessary mathematical work will be mainly supplementary. 
If the subject has to be included wholly in the mathematical 
course, it is not likely to have any great value. Mathematical 
astronomy which is not based upon personal observations 

*This chapter should be read in conjunction with Chapter XXVI of Science 
Teaching. 

(E291) 33 



498 CRAFTSMANSHIP IN MATHEMATICS 

of any kind, with the telescope at least, if not with the 
spectroscope, is not likely to have much reality. 

Elementary Work 

A certain amount of introductory astronomy will neces- 
sarily be included in a school geography course. For 
instance: 

1. The earth as a globe travelling round the sun and 
spinning all the time on its own axis inclined 661 to the 
plane of the ecliptic, i.e. the plane of its path round the sun. 

2. The consequences of these movements: day and night, 
the seasons. 

3. The moon as a globe spinning on its own axis once a 
month, and travelling round the earth once a month, in a plane 
slightly inclined to the plane of the ecliptic. Phases of the 
moon. 

4. Eclipses: comparative rarity of the phenomenon the 
result of the inclination of the orbits of the earth and moon. 

5. Fixing positions on the earth's surface. Latitude and 
longitude. Elementary notions of map projection. 

Older pupils who have done a fair amount of geometry, 
especially geometry of the sphere, have no difficulty in under- 
standing these things from descriptions and diagrams. But 
younger pupils require more help, otherwise they cannot visu- 
alize the phenomena, they remain puzzled, and their written 
answers to questions are seldom satisfactory. 

If an orrery is available, there is little difficulty, but more 
often than not the teacher has to manage with improvised 
models, perhaps a mounted globe to represent the earth, and 
painted wooden balls to represent the sun and moon. Per- 
sonally I prefer to use a large porcelain globe (the kind used 
with the old-fashioned paraffin lamps) to represent the sun, 
the globe being fixed in position a foot or so above the centre 
of the table, and illuminated from the inside by the most 
powerful electric light available, the room being otherwise in 



ASTRONOMY 499 

darkness. This makes an admirable sun, and gives a sharply 
defined shadow. The earth may be represented by a small 
wooden ball painted white, with a knitting-needle thrust 
through its centre to represent the axis, and with black circles 
to represent the equator and the 23^ and 66^ parallels, the 
ball being mounted so that its centre is the same height 
above the table as is the centre of the sun, and the axis being 
inclined at 66-. About one-half the " earth " is now brilliantly 
illuminated, and the other half is in shade. If the earth is 
moved round in its orbit, the successive positions of its axis 
maintaining a constant parallelism, the meaning of (i) day 
and night and their varying length in different parts of the 
world, and (ii) the seasons, may be made clear in a few 
sentences. If more serious work is to be done later, it is 
particularly necessary that the plane of the ecliptic should 
be clearly visualized, and this is easily done if the sun and the 
earth are supposed to be half immersed in water, the surface 
of the water representing the plane of the ecliptic. Make 
the pupils see clearly that half the earth's equator is always 
above, and the other half always below, this plane. 

The phases of the moon are best taught by ignoring the 
model of the earth for the time being and considering models 
of the sun and moon alone. Let the laboratory sun illuminate 
a painted ball, to represent the moon; let the pupils move 
round this ball, from a position where they see the non- 
illuminated half to the position where they see the fully- 
illuminated half. One " phase " after another comes into 
view, and further teaching is unnecessary. Now put the 
"earth" in position, and show how the earth may get between 
the sun and the moon, and prevent the sun from shining 
on the moon; and how the moon may get between the earth 
and the sun, and prevent our seeing the sun. And thus we 
come to eclipses. 

The first essential in teaching eclipses is to make pupils 
realize that a cone of shadow is a thing of three dimensions. 
Let the school sun cast the shadow of the much smaller 
school earth. The whole classroom remains brilliantly lighted 



500 CRAFTSMANSHIP IN MATHEMATICS 

save for a cone of darkness on the far side of the earth (we 
ignore all other objects in the room), and the shape and 
size of this cone is easily demonstrated by holding a screen 
at varying distances behind the earth. With a second ball 
to represent the moon, correct notions of total, annular, and 
partial eclipses may be readily given. It is quite easy to 
show why eclipses are comparatively rare phenomena by 
making the moon move round in an orbit inclined to the 
earth 's orbit. 

More Advanced Work 

A Sixth Form ought to carry the subject very much 
farther than the elementary aspects of it commonly included 
in a geography course, but the business of the mathematical 
teacher is not to give astronomy lectures in the wider sense 
but to teach boys to solve those problems which are suggested 
by the results of actual observation; for instance, the problem 
of fixing the positions of the stars by means of their co- 
ordinates, the related question of the diurnal revolution of 
the heavens, the daily movements of the sun and moon, 
the calculation of times of rising and setting, nautical problems 
of determining latitude and longitude, dialling problems. 

Facts must not be confused with hypotheses. Thus the 
earth's daily rotation on its axis and its annual revolution 
round the sun are mere hypotheses, invented to account for 
facts of observation. The mathematical teacher is concerned 
with the face value of the facts observed. According to that 
face value, the stars move round the sky daily, and the sun 
and moon move amongst them. Any attempt to provide 
a theory of stellar movements must be preceded by an exact 
determination of the facts as they appear. 

Quite low down the school the boys ought to have been 
made familiar with the globe (a blackboard surface is very 
useful) and a cardboard horizon fitting over it. And in the 
very early stages of geometry they will have been introduced 
to the theodolite, and will have been taught to measure 
altitudes and azimuths (though perhaps the term azimuth 



ASTRONOMY 501 

has not been used). The theodolite may have been made in 
the school workshop, and a mere cardboard tube used instead 
of a telescope. But higher up the school an instrument 
designed for fairly accurate measurements should be used, 
and nowadays a good one may be purchased for a few pounds. 
Even Fourth Form boys can be taught to measure the azimuth 
and altitude of a given star as it appears to an observer at 
a given moment. It is easy and interesting work and they 
like it, though some of them seem to need repeated help 
with the setting up and initial adjustment of the instrument. 
I have known boys of 9 or 10 readily pick out the better 
known constellations, and such stars as the Pole Star, Vega, 
Capella, Sirius, and the Pleiades. This kind of observation work 
ought to be included in every Nature Study course. It creates 
an early interest that becomes permanent, and such basic 
facts are very useful for future mathematical work. 

A school lucky enough to have a small observatory of 
its own will have an altazimuth (a theodolite is virtually 
a portable altazimuth), so that azimuths and altitudes (or 
zenith distances) may readily be found. An equatorial may 
also be available. If not, the altazimuth should be of such 
a kind that its telescope can be mounted equatorially when 
required. Then the boys can take Declinations and Right 
Ascensions, and become familiar with the celestial equator 
as well as with the celestial pole, and they will then soon 
look upon the rotating northern celestial hemisphere as an 
old familiar friend. Once they feel this familiarity, the 
making of reasonably accurate observations is child's play 
and the mathematics involved is not difficult. The sidereal 
clock and sidereal time are also easily mastered. 

The solution of the common problem of determining the 
altitude and azimuth of a star when the hour- angle and 
declination are given (or vice versa) is an easy case of the 
solution of a spherical triangle, and should be familiar. 

The sun-dial cannot profitably be taken up until the 
Sixth, and not even then unless the boys have been well 
grounded in the geometry of the sphere and its circles. The 



502 CRAFTSMANSHIP IN MATHEMATICS 

geometrical method of graduating the dial (to be fixed either 
horizontally or on a south wall) is simple enough if the 
elementary geometry of the sphere has been mastered. The 
boys must be able to see that the key to the whole thing lies in 
the fact that the edge of the gnomon is parallel to the earth's 
axis and is therefore pointing in the direction of the celestial 
pole. If about this they are vague, the whole thing is vague. 

There is no better way of introducing the young observer 
to the knowledge of the law of the sun's rotation than by 
leading him to see that, if a dial be so placed that the style 
(the edge of the gnomon) is parallel to the axis of the rotating 
celestial hemisphere, the shadow of the style will at all seasons 
of the year move uniformly over the receiving surface at the 
rate of 15 an hour. 

The graduation of a sun-dial to be placed on a vertical 
wall is not difficult, but it is a good little puzzle for testing a 
boy's knowledge of the sphere and his powers of visualizing 
the true geometrical relations of the parts of a rather com- 
plicated figure. 

Mathematical problems in astronomy are, of course, un- 
limited, but in school there is no time to touch upon more 
than the bare fundamentals. 

Whitaker's Almanack is a mine of useful data for problem 
purposes. 

Stellar Astronomy 

The main interest of astronomers, and indeed that of the 
general public, is now concerned with the stars and nebulae 
rather than with the solar system." With the main facts of 
the solar system every boy should be made familiar; but 
stellar astronomy is more difficult, the greater part of the 
available evidence being merely of an inferential character. 
In a very large measure we have to deal with probabilities, 
not certainties. 

The astronomer's principal instruments are the telescope 
(mounted in different ways according to the work to be done), 
the spectroscope, the camera, and the interferometer. The 



ASTRONOMY 503 

last-named is outside possible school practice, so is the 
camera. But the spectroscope is now in common use in 
schools, and as it ranks next to the telescope in the work 
of an observatory, its uses should be taught thoroughly. 

A course of instruction may be expected to include the 
following: 

1. Spectrum analysis. Displacement of lines: the causes; 
difficulty of interpretation; distance and speed effects con- 
sidered separately. 

2. The galactic system of stars. 

3. The extra-galactic system: stars and nebulae. 

4. Stellar spectra. Interpretation of photographs. 

5. Stellar magnitudes, movements, velocities, distances, 
temperatures; how determined. 

6. Theories of stellar structure: for instance, (i) Edding- 
ton's, (ii) Jeans'. 

7. Solar radiation. Energy and temperature of sun. 
Poincare's theorem. 

8. Stellar radiation and cosmic radiation generally. Hoff- 
mann's determination of the sun's contribution to the total 
cosmic ultra-radiation; inferences therefrom. Hess's views. 

9. Relativity. General outline. Einstein's proposed tests. 
Confirmation of the tests and final acceptance of the theory. 

10. Modern cosmologies: (i) Einstein's, (ii) De Sitter's. 
Do they clash? Lemaitre's views how an Einstein universe 
may expand to a De Sitter universe. 

11. Rival theories as to the future of the universe. British 
physicists' views of a universe slowly running down to a state 
of thermodynamic equilibrium. Millikan's views of a universe 
being continually rebuilt. Evidence pro and con. 

How much of this work will be done by the mathe- 
matical teacher? His task will probably be concerned mainly 
with two things: (i) some easy but extremely interesting 
arithmetic; (ii) the very difficult subject of Relativity. 

Mathematical teachers differ in opinion as to the wisdom 
(or folly) of introducing relativity in a Sixth Form course. 



504 CRAFTSMANSHIP IN MATHEMATICS 

But in view of the far-reaching, indeed fundamental, changes 
that the subject is bringing about in the whole domain of 
physics, it seems desirable that an attempt should be made 
to give Sixth Form specialists at least an outline of the subject. 
After all, the " special " theory of relativity is easily taught, 
and, this done, the much more difficult " general " theory 
may be so far touched upon that the final results of the theory 
may be fairly well understood by the abler boys. Professor 
Rice's and Mr. DurelPs little books may be followed up 
by Einstein's own elementary book, and his by Nunn's 
Relativity and Gravitation, which is by far the best book 
on the subject from the teacher's point of view.* 

The arithmetic of stellar astronomy deals with numbers 
so vast that it is likely to deceive all but the trained mathe- 
matician. How, for instance, may we bring home to a boy 
the real significance of the following: 

1. The sun is losing weight by radiation at the rate of 
1-31 . 10 14 tons a year, yet 2 . 10 years ago it was only 
1-00013 times its present weight. 

2. Weight of sun 2 . 10 33 grammes. 

3. Temperature of interior of sun = 4 . 10 8 degrees. 

4. Number of stars in galactic system = 4 . 10 11 . 

5. The 2,000,000 extra -galactic nebulae each contain 
enough matter to make 2 . 10 9 stars, that is 4 . 10 15 stars in all. 

6. The extra-galactic nebulae are at an average distance 
away of 140 million light-years (1 light-year = 6 . 10 12 miles) 
and their average distance apart from each other is of the 
order of 2 million light-years. 

7. Radius of universe is perhaps 2000 million light-years 

= 2 . 10 9 X 6 . 10 12 miles 
= 1-2 . 10 22 miles. 

We shall refer to this subject again in a later Chapter. 

Be consistent when using the terms " world ", " uni- 
verse ", " cosmos ", " space ", " ether ", " space-time ". It 

* For detailed suggestions see Chapter XXXII of Science Teaching. 



ASTRONOMY 505 

is probably sufficient to tell a boy that the matter- containing 
universe, no matter how large, is itself within a limitless void. 
Do not let him think that the mathematician's convenient 
and necessary fiction " space-time " is any sort of glorified 
Christmas pudding mixture. The mathematical partnership 
is purely formal. Distinguish between an infinite void and 
a limited wave-carrying matter-containing universe. 

In his address to the Mathematical Association, January, 
1931, Sir Arthur Eddington said: "About every 1,500,000,000 
years the universe will double its radius and its size will 
go on expanding in this way in Geometrical Progression for 
ever." A rude boy might ask some very awkward questions 
on this point, and carry his teacher backwards as well as 
forwards in limitless time. It is of no use merely to go back 
to an assumed initial state of equilibrium. The boy is certain 
to say, and before that? 

Books to consult: 

In selecting books on Astronomy, don't forget some of the older 
writers, e.g. Herschel, Proctor, Lockyer, Ball. Eddington 's, Jeans', 
and Turner's books should be known to all teachers of mathematics. 
Barlow and Bryan's Elementary Mathematical Astronomy is very 
useful. From the teacher's point of view, Sir Richard Gregory's 
books take quite the first place. Consult also Dingle's Astrophysics. 

Readers who are specially interested in Relativity should read 
Dr. John Dougall's searchingly critical article in Vol. X of the 
Philosophical Magazine, pp. 81-100. 



503 CRAFTSMANSHIP IN MATHEMATICS 



CHAPTER XXXIV 

Geometrical Optics 

Present Methods of Teaching often Criticized 

We include this subject because it quite properly belongs 
to mathematics as well as to physics. 

Probably no part of the teaching of mathematics or of 
physics is so severely criticized as the teaching of optics, no 
matter whether the subject is taught by the mathematics teacher 
or by the physics teacher. That there is an urgent need for 
some reform will be readily admitted from the discussion 
on " The Teaching of Geometrical Optics " that took place 
on April 26, 1929, reported fully on pp. 258-340 in No. 229 
of the Proceedings of the Physical Society. Papers were read 
by a number of persons interested in optics, including several 
Public School and University teachers and representatives 
of the optical industry. A few of the teachers tried to defend 
the present system, though not very successfully. The conflict 
of opinion centred largely (1), round the place to be given and 
the purpose to be assigned, in a teaching course of optics, to 
the reciprocal equation (1/u -\~ I/v I/f), and (2), round the 
question of " rays or waves ". My own quite definite con- 
clusion from the discussion was that the best way of teaching 
the subject is to begin with elementary physical optics in 
Forms IV and V, and to defer geometrical optics until Form VI. 

Several of the critics found fault with the present system 
because it fails to supply a sufficient practical knowledge 
of optical instruments and their performance; because pupils 
by the end of their course in optics have done little more than 
devote their time to elementary algebra and geometrical 
diagrams which have but a very slender relation to the subject 
under consideration; because, in short, the utility of the subject 
is extremely meagre. 



GEOMETRICAL OPTICS 507 

My own main criticism takes another direction that 
the mathematics and the theory of the subject at present 
tend to take too early a place in the teaching course, inasmuch 
as the physical phenomena underlying the mathematics and 
the theoretical arguments have not been studied, the arguments, 
therefore, having no real significance. 

Rays or Waves? 

Hitherto the " ray " method of teaching has been almost 
universal in our schools, but the mathematics has been 
too much divorced from experiment and its real significance 
has been ill understood. In the discussion already referred 
to, the method was defended mainly because of its simplicity, 
not because of its practical utility. The protagonist of the 
wave or curvature method was Dr. Drysdale, for many 
years head of the optical department at the Northampton 
Institute, London. He advocated the method on the grounds 
(amongst others) that (1) it simplifies the teaching; (2) it 
harmonizes the teaching of science with optical practice; 
and (3) it leads naturally to higher physical optics. The real 
advantage of the method seems to be that it places the whole 
of optical teaching on a physical basis, and leads naturally 
to the study of interference, diffraction, and polarization. 
Two well-known elementary books developing the subject 
on a wave basis are those of Mr. W. E. Cross and Mr. C. G. 
Vernon. 

Whichever method is used, the teacher should be quite 
frank in stating that energy can be radiated in two forms, 
corpuscles and waves. Both forms are easily illustrated experi- 
mentally. For example, replicas of diffraction gratings (if 
gratings themselves are too expensive to buy) are suitable 
for illustrating the periodic character of light. In fact, the 
periodic character of light must be experimentally demon- 
strated in some way before the curvature method can logically 
be introduced, and this means a preliminary study of the 
velocity of light. 



508 CRAFTSMANSHIP IN MATHEMATICS 

It is a good thing to teach both methods, and to teach 
them more or less in parallel. A ray may, for instance, be 
looked upon as a line representing an element of the wave- 
front, or as a normal to the wave-surface; or the wave-front 
may be traced as a series of arcs after the rays have been drawn 
graphically. 

The best defence of the wave method is that the whole 
of physics is, fundamentally, a study of wave systems, and 
it is therefore difficult to justify the picking out of one branch 
and treating it on an entirely different basis. But the ob- 
jection to the ray method largely disappears if the ray be 
thought of as an element of a wave, and to the lens designer 
the ray is the all-important thing. 

Theories of Light 

The whole question turns largely on an acceptable theory 
of light. But whose theory? Newton's? Fresnel's? Young's? 
Maxwell's? Planck's? de Broglie's? 

Newton's corpuscular theory failed to account for certain 
observed facts. The wave theory which superseded it was 
also found to be defective, and to eliminate these defects the 
" quantum " theory has been devised. The new theory has 
shown that Newton was not wholly wrong in regarding light 
as corpuscular, for that theory is based on the experimental 
fact that a beam of light may be considered to be broken up 
into discrete units called " light-quanta " or " photons ", 
" with almost the definiteness with which a shower of rain 
may be broken up into drops of water, or a gas into separate 
molecules ". At the same time, the light preserves its undu- 
latory character. Each photon has associated with it a perfectly 
definite quantity of the nature of a wave-length. 

There seems to be no doubt at all that radiation of all 
kinds can appear now as waves, now as particles. But the 
fundamental units of matter, electrons and protons, can also 
appear now as waves, now as particles. In many circumstances 
the behaviour of an electron or proton is found to be too 



GEOMETRICAL OPTICS 509 

complex to permit of explanation as the motion of a mere 
particle, and accordingly physicists have tried to interpret it 
as the behaviour of a group of waves, and in so doing have 
founded the branch of mathematical physics known as " wave- 
mechanics ". 

In fact it may be fairly said that no single satisfactory 
theory of light exists to-day. The electromagnetic theory 
carries us a long way, but in its classical form it is quite 
inadequate to carry us the whole way. The powerful methods 
devised by Hamilton in geometrical * mechanics and geome- 
trical optics are being used to found a wave-mechanics bearing 
to geometrical mechanics a relation similar to that which 
wave-optics bears to geometrical optics. The quasi light- 
particles emerge from this mechanics more or less naturally, 
so that we are practically back to Newton and working on 
Newton's lines. The two views are blended; neither is 
destroyed. 

Geometrical optics is as worthy of serious study as geome- 
trical mechanics. Each is the limiting form when A ~> 0, 
and for many purposes this limiting mathematical form is 
not only entirely sufficient but it is vastly simpler, mathemati- 
cally, than the general wave-form, whether in optics or in 
mechanics. What is not worthy of study (at all events as 
physics) is the type of question often set, the solution of which 
depends wholly on some mathematical trick. Large numbers 
of these are found in such favourite old books as Tait and 
Steele's Dynamics, or Parkinson's Optics, or Heath's Geometrical 
Optics. Such problems are possibly good as training material 
in mathematics, but for the display of mathematical talent 
there is an abundance of excellent material that is, in itself, 
valuable in physics also. 

* I.e. Newtonian. 



510 CRAFTSMANSHIP IN MATHEMATICS 



The Teacher of Optics 

Should optics be taught by the mathematics teacher or 
by the physics teacher? Admittedly a mathematics teacher 
who has had no training in physics is not likely to be able 
to appreciate the natural powers and limitations of optical 
instruments, or to grasp the significance of certain matters 
in optical theory. Admittedly, too, a physics teacher with 
no special knowledge of mathematics will be out of his depth 
in the Sixth Form where, in optics, mathematical considerations 
count for almost everything, though he will be easily able to 
cope with the first considerations of the reciprocal equation, 
which, after all, is essentially a natural development of fives- 
court and billiard-table geometry. There is thus very little 
doubt about the answer to our question. The physics teacher 
should be responsible for the physical optics in Forms IV and 
V, and the mathematics teacher for the geometrical optics 
to be done in VI. By geometrical optics is here meant the 
really serious mathematical work that should follow the physical 
work, work that is partly revisionary but mainly supplementary. 
The higher physical work in VI will, however, still have to be 
taken by the physics teacher. 



Suggested Elementary Course: Mainly Physics 

This elementary course is intended to be mainly experi- 
mental and to be done in the laboratory, all consideration 
of the theory of aberration being excluded. Let all serious 
mathematics and theoretical developments be postponed to VI. 

If the wave method is adopted, wave motion and its sig- 
nificance will naturally be taught first. Of the many wave- 
producing machines in the market, select one, and see that 
the boys really understand what it teaches. The propagation 
of transverse waves may be shown by a ripple tank, illuminated 
stroboscopically, so that the apparent rate of propagation may 



GEOMETRICAL OPTICS 511 

be slowed down. Carry out practical work with real beams of 
light, not by pin and parallax methods. The sunbeam offers 
a concrete starting-point. 

Devote a lesson or two to showing how fallible the eye 
is as a measuring instrument, and why, therefore, instrumental 
aids are necessary. Devise experiments to show the limited 
power of the eye in unaided vision, and show the capacity 
of the eye for distinguishing detail under different conditions 
of illumination and size of aperture. 

Make beginners familiar with the construction and use 
of optical instruments the telescope, the microscope, and 
photographic lens. When a boy handles optical instruments, 
and learns to adjust, to test, and to use them, he acquires 
knowledge of their potentialities and limitations; and he 
also becomes acquainted with the language of the subject. 
Throughout the course keep in mind elementary notions 
both of physiological optics and of the psychology of vision; 
also that the eye as an optical instrument is very imperfect, 
deceptive, and inconstant. Teach beginners when using 
optical instruments the importance of correct illumination; 
and the uselessness of increasing magnification beyond the 
value suitable for the aperture actually effective in the experi- 
ment. Show that the apparent brightness of an extended object 
cannot be increased by optical means; the moon looks no 
brighter through a telescope. 

The key to refraction is, of course, the mere retardation 
of velocity in a denser medium, and the boys must under- 
stand clearly that a refractive index is simply a velocity ratio. 
The slewing round of the wave-front must be understood 
to be just a natural and inevitable consequence of any such 
retardation and to be applicable universally and not merely 
in connexion with light. The trundling of a garden roller 
across a smooth lawn to a rough gravel drive affords a service- 
able illustration. If the direction of motion across the lawn 
is normal to the line of separation between grass and gravel, 
there is merely retarded velocity; if oblique, there is a slew- 
ing round as well. 



512 CRAFTSMANSHIP IN MATHEMATICS 

Suggested topics: 

1. Nature and propagation of light. 

2. Waves: motion, length, amplitude, frequency, velo- 
city. 

3. Illumination. Photometry, especially the measurement 
of illumination by daylight photometer.* 

4. Experiments in brightness, colour, persistence of vision, 
fatigue, glare. 

5. Reflection and refraction. Concept of the ray as a 
line representing the direction of movement of an element 
of the wave-front. The use of rays in optical diagrams. 
Huygens' principle. 

6. Function of lenses; imprinting of curvature. 

7. Interference, diffraction, polarization. 

8. The spectrum; the spectroscope. 

9. The spectrometer: first considerations. 

10. The beginnings of mathematics; the reciprocal equation 
as a convenient memorandum for elementary work at the 
optical bench. 

11. Inverse square law; the unit standard source of light, 
the unit of luminous flux, the unit of illumination, and their 
interrelations. 

Suggested Advanced Course: Largely Mathematics 

Whatever books on geometrical optics teachers use, 
especially if they are old favourites like Parkinson and Heath, 
it is a good plan either to compare these with modern standard 
works on the technical side of the subject, or to discuss them 
with a friend acquainted with the optical industry. The im- 
portant thing is to find out if the principles laid down in a book 
will work. 

An examiner reports that at a recent university examination he set a simple 
question on the measurement of daylight illumination. Hardly any of the 240 
candidates gave a complete answer. A common plan was to balance sunlight 
against an electric lamp, using, say, a grease spot, assume the sun to be 93,000,000 
miles away, assume the inverse square law, and to calculate the candle-power of the 
sun! 



GEOMETRICAL OPTICS 513 

There is no excuse whatever for teaching the subject 
by methods that are out of harmony with applied optics. 
Young computers who are taken on at optical works often 
find to their disgust that their school and textbook knowledge 
is valueless, and they have to be taught anew by technical 
experts. 

The commonest mistake in optical teaching is due to 
the misuse or to the misunderstanding of the sign convention 
and notation. This is unaccountable, as the convention 
is the result of international agreement. In the optical dis- 
cussion already referred to, an examiner said that a year or 
two ago he marked 250 scripts in the Higher Certificate 
examination, the candidates having been taught in schools 
in different parts of the country. In the Light paper was a 
simple question on a lens, and 247 of the candidates attempted 
it, but only 7 of the 247 obtained the correct result. Such a 
record of muddleheadedness is utterly inexcusable. 

Remember that the basis of all lens work calculation should 
be the deviation in a ray at each surface. Suppose that a ray 




which diverges from the point A at, say, 15 is to emerge from 
the lens system, S, parallel to the axis. Since the whole 
deviation is to be 15, and if there are, say, two surfaces, 
are the two partial deviations to be 7| each, or in some 
other proportion? What are the criteria for what is best? 
What are the aberrations? And so on. 

After a few simple calculations on a simple lens for actual 
wide-angle cases, the boys will soon find that the rule given 
by Parkinson for the relative radii of the surfaces is by no 

(B291) 34 



514 CRAFTSMANSHIP IN MATHEMATICS 

means always right; it is only right when a ^ 10 (about), 
while in many lenses a is very much greater. 

Wave optics must not, of course, be forgotten. For instance, 
the wave equation and its simple solution should be included. 

Suggested topics: 

1. Geometrical optics: the reciprocal equation more fully 
considered. " Wave " proofs and " ray " proofs compared. 

2. The dioptre, sagitta (sag), focal power. Show that the 
curvature of a wave-front or surface is measured by the reciprocal 
of the radius; the surface with a radius of 1 m. is chosen 
as a standard. R dioptres = l/^ met res- Exhibit a curve of 1 m. 
radius so that the curvature may be visualized. Point out that 
for a chord of 8-95 cm. the curvature in dioptres is represented 
by the sag in mm. The application of Euclid, III, 35, to the 
sag. The dioptre spherometer. 

3. The ideal lens contrasted with the actual lens. (The 
solution of problems arising out of actual lenses will in general 
be too difficult.) 

4. Lenses; spectacles. How the optician is concerned 
with the forms of lenses as well as with their power. 

5. Combination of lenses with prisms to correct defects 
of convergence in the eyes. 

6. Thin lenses in contact. 

7. Lens combinations. 

8. Axial displacement. 

9. Chromatic aberration. 

10. Spherical aberration: the disc of confusion. 

11. Astigmatism, coma, distortion. 

12. Photometry further considered. How the distance of 
star clusters and spiral nebulae have been determined by the 
measurement of the apparent brightness of Cepheid variables 
contained therein. 

13. Modern instruments; the telephoto lens, range- 
finders, prism binoculars, kinema projectors. 

14. The more elementary considerations of such subjects 
as defects of images, collineation between object space and 
image space, the optical sine theory, design of instruments. 



GEOMETRICAL OPTICS 515 

The correction of aberrations by calculation will, in 
general, be too difficult; so will the higher order of aberrations 
considered by ray-tracing, though some notion of ray-tracing 
should certainly be given. The general theory of lenses will 
also be too difficult. In short, a good deal of this work is 
more suitable for the university than for the school. Much 
will depend upon the close collaboration of the mathematics 
and physics staff. The two aspects of the subjects must be 
considered together. 

The real value of mathematical work in optics lies in the 
discovery of the general principles underlying the actual 
behaviour of real optical systems, as contrasted with the imagined 
behaviour of ideally perfect systems. 

Technical Optics 

Few teachers are familiar with technical optics. Very 
few have seen even the ordinary operation of grinding a 
lens. As for the designing of lenses for special purposes, 
or the art of producing optical glass, it is known to very 
few persons indeed. Few teachers realize that for ordinary 
industrial purposes the index of a glass is not considered 
known unless its value is obtained to the fourth decimal 
place, and for dispersion to the fifth. 

Formerly when an optical system had been designed, the 
material prepared by the designer was handed over to a 
number of computers expert in the use of logarithmic tables. 
But calculating machines are now used, to the operators of 
which the computation of the elements, individually and in 
combination, of the new optical system is entrusted. These 
operators need have no special mathematical equipment, 
other than that of a common knowledge of simple trigo- 
nometrical expressions. Particular rays are traced step by 
step through surface after surface for the purpose of de- 
termining at various stages the longitudinal and transverse 
aberrations. These values are assessed by the skilled com- 
puter, who decides at what particular part of the system a 



516 CRAFTSMANSHIP IN MATHEMATICS 

modification can best be effected. His special skill is practical, 
the outcome of active practice in the industry itself. It 
involves, above all, good judgment in the balancing of 
one type of aberration against another, for no optical 
system can be free from all kinds of aberration. Consider 
the amount of work involved in the computation of the 
optical system of a typical submarine periscope. Altogether 
the number of separate operations is something like 40,000, 
the mere recording of which would fill a book of some 250 
pages. 

Of course all this sort of work is entirely outside anything 
that can be done in school, but if a teacher himself is entirely 
ignorant of it, how can he help making his subject unreal, 
and talking about it in a foreign tongue? 

The Sign Convention 

Many of the difficulties underlying the teaching of ele- 
mentary optics in the past have arisen because teachers 
have adopted different practices in the use of signs. The 
following diagram shows the sign convention that has been 



INCIDENT LIGHT ^ 



agreed upon by the principal optical authorities in the country. 

Books to consult: 

1. Optics, W. E. Cross. 

2. Light, C. G. Vernon. 

3. The Theory of Light (new ed.;, T. Preston. 

4. Introduction to the Theory of Optics, A. Schuster. 

5. Experimental Optics, C. F. C. Searle. 

6. Practical Optics, B. K. Johnson. 

7. Theory of Optics, P. Drude (trans, by Mann and Millikan). 

8. Optics, Muller-Pouillet, 3rd ed. 



GEOMETRICAL OPTICS 517 

9. Principles and Methods of Geometrical Optics, J. P. C. Southall. 

10. Optical Measuring Instruments, Prof. L. C. Martin. 

11. Optical Designing and Computing, Prof. Conrady. 

12. Proceedings of the Physical Society, No. 229; the papers by 
Mr. T. Smith, Dr. Searle, Dr. Drysdale, Mr. C. G. Vernon, Captain 
T. Y. Baker, are all very instructive. 

The reader may usefully refer to the memorandum prepared, in 
January, 1931, by the Council of the British Optical Instrument 
Manufacturers' Association. The facts adduced definitely establish 
the pre-eminence of the British position in the optical industry. 
The tests effected in the National Physical Laboratory are alone 
sufficient to make that clear. 



CHAPTER XXXV 

Map Projection 

Developable and non -Developable Surfaces 

It is the geography teacher's business to show how maps 
can be outlined on the particular graticule system prepared 
for him by the mathematician. This graticule system a 
gridiron or lattice-work system of parallels and meridians 
is in its very essence mathematical and should be included 
in every school mathematical course. 

Fundamental principles of projection will already have 
been taught in the lessons on geometry. The principles of 
orthographic projection, including so-called " plans and 
elevations ", should have been taught thoroughly. It is 
just an affair of parallels and perpendiculars, and thence to 
the idea of parallel rays of light from an indefinitely distant 
source is but a step. 

The geometry of the sphere should also be known 
thoroughly; for instance, that the area of a circle is ?rR 2 ; 
of a sphere, TrD 2 and therefore 4 times one of its great 



518 CRAFTSMANSHIP IN MATHEMATICS 

circles; of a hemisphere, twice that of its great circle; and 
that the volume of a sphere is 7rD 3 /6. 

It should be realized that when we look at a sphere we 
cannot see the whole of a half of it. The portion of the 
visible surface is that encircled by a tangent cone with its 
apex at the eye (we neglect binocular vision). A photograph 
of a geographical globe would necessarily give a picture of 
rather Jess than a hemisphere. 

Developable surfaces is another subject that should have 
been taught. A paper model of a cube, prism, or pyramid 
can be slit open along some of its edges and laid out on the 
flat, in other words, " developed ". A cylinder or cone 
can be similarly treated. On a cylinder or cone straight lines 
can be drawn in certain directions; if the cylinder or the 
cone is lying on the table, the line of contact with the table 
is one such straight line. But a spherical surface is altogether 
of a different type; no straight line can be drawn upon it; 
it cannot be developed. A sphere touches a plane at a 
point. We cannot cover a sphere with a sheet of paper as 
we can a cylinder or cone. 

Now the earth is approximately spherical, and any correctly 
drawn map is part of that spherical surface. An atlas of 
true maps would consist of spherical segments, not flat 
sheets. Such an atlas has been made in metal, but it is clumsy 
to use and is expensive. For convenience we draw our maps 
on the flat, and thus they are all wrong. A map of England 
drawn to scale on the surface of an orange would be very 
small but large enough for a needle to be thrust through 
the orange along a chord from Bournemouth to Berwick. 
A perfectly straight tunnel driven between these towns 
would pass under Birmingham 4 miles below the surface. 
If a map of Europe be sketched to scale on a hollow india- 
rubber ball, and that portion of the ball be cut out, the portion 
has to be stretched a great deal to lie flat, and thus parts 
of the map are greatly distorted. 

Evidently no map can be drawn on a flat surface accurately. 
How do map-makers set to work? 



MAP PROJECTION 519 

If we examine an ordinary geographical globe, we see 
the equator, the north and south poles, meridians of longitude 
running from pole to pole, and diminishing circles of latitude 
running " parallel " to the equator. And on this network of 
lines we see a true map of the world. 

To draw a map, we first draw a network of lines corre- 
sponding as nearly as possible to those on the surface of the 
globe, though they are bound to differ very considerably 
from the originals. The network once drawn, we put into 
each little compartment, as accurately as we can, the corre- 
sponding bit of map on the globe. The real trouble is to 
draw the network. 

An examination of an atlas shows that the various net- 
works differ much in appearance. Sometimes one or both 
sets of lines are straight, sometimes curved, and the cur- 
vature seems to vary in all sorts of ways. Why? This we 
must try to find out. 

In an ordinary plan drawn to scale, say of a house or of 
a town, we have the simplest form of projection, called the 
" orthographic ". To every point in the original corresponds 
a definite point in the drawing, and the spatial relations 
between the points are faithfully reproduced; only the scale 
is changed. 

But in a map, the relations may all be changed. There 
will, however, still be a systematic one-to-one correspondence 
of points. Some sort of general resemblance to the original 
may always be easily detected, though there is certain to be 
distortion of form, or inequality in area, or both. 

The map-maker is bound to sacrifice something. If he 
is making a map for teaching geography, he tries to represent 
correctly the relative sizes of land and sea areas and thus 
provides an equal-area projection. If he is making a map for 
a navigator, he tries to show correct directions, and does 
not trouble much about size. Or he may be concerned 
mainly with correct shapes, and not much with sizes and 
directions. Hence he has contrived projections for different 
purposes. He has to be content to represent a portion of 



520 CRAFTSMANSHIP IN MATHEMATICS 

the earth's surface accurately in certain respects and to 
let other considerations go. 

The plan adopted is to project the curved lines from the 
globe on to (1) a plane surface, or (2) a developable surface 
(cylinder or cone). 

Some projections are readily effected geometrically; they 
are easy to draw and to understand. Other projections are 
not strictly geometrical: they are compromises, effected for 
some particular purpose, and are often called transformations. 
In these cases point-to-point correspondence is determined 
merely by formulae which express the position of each point 
on the plane of the projection in terms of the position of the 
point on the spherical surface to which it corresponds. 

Projection Shadows 

It is possible to obtain geometrical projections by casting 
shadows. A light is placed in a suitable position, and a pencil 
outline of the shadow of the globe is traced on a conveniently 
placed plane. This done, it is easy to see how a better pro- 
jection may be made with ruler and compasses. 

Of course if we use a solid globe the shadow will be 
merely a black circle. We require a hollow translucent 
globe, with the meridians and parallels painted black on the 
surface, and a strong light inside. If the globe is fixed near 
a sheet of white paper on the wall or on the table, the shadows 
of some of the meridians and parallels will be cast on the 
paper, and those fairly near the globe will be clear enough 
to be pencilled over. A large white porcelain globe used for 
gas and electric lighting answers the purpose well. 

When teaching 40 years ago, I found that a better plan 
was to use a spherical wire cage instead of a globe, made 
something after the pattern of the old-fashioned wire pro- 
tectors of naked gas-flames in factories. Such a cage 2' or 
2' 6" in diameter is easily made in the school workshop. 
It is merely a question of bending wire and soldering a number 
of joints. For the equator, a rather stouter wire should be 



MAP PROJECTION 



521 



used than for the other circles. The meridians are best 
not made of complete circles but of rather less than half 
circles, fastened into a ring 4" or 5" in diameter, after the 
manner of the ribs at the top of an umbrella. It is true that 
the actual north and south poles will be missing but this 
cannot be helped, the crossing of 12 wire circles at a common 
point not being practicable. The meridians and parallels 
may be placed at 15 intervals. Two half-cages are also 
desirable, one with a pole at its centre, one with a point on 






Fig. 275 

the equator at its centre. The three should be mounted on 
suitable stands, in order that, in use, they may easily be 
kept in a fixed position. 

The main difficulty is the provision of a suitable light. 
Theoretically we require the light to be concentrated at a 
point. As this is impossible, we use a small electric bulb, 
porcelain or similar material, with the most powerful light 
obtainable. A darkened room is, of course, necessary. 



Main Types of Projection 

The principal projections may be grouped under six 

main heads: (1) zenithal or azimuthal; (2) globular; (3) 

conical; (4) cylindrical; (5) sinusoidal; (6) elliptical. Of 
most of these there are various modifications. 



522 CRAFTSMANSHIP IN MATHEMATICS 

(1) Zenithal or Azimuthal Projection 

This projection derives its two names from the facts 

(1) the map is symmetrical about its central point, just as 
the stellar vault is symmetrical about the zenith of the observer; 

(2) the projection preserves the azirmiths of distances measured 
from the map's centre. 

There are three distinct types of this projection: (1) 
orthographic; (2) stereographies (3) gnomonic. (See fig. 276.) 

1. Orthographic. This is simply an affair of perpen- 
diculars and parallels. As we cannot obtain parallel rays 
by artificial light, we must use the sunlight at some con- 
venient hour. Let the paper prepared to receive the shadow 
be placed at right angles to the direction of the sun's rays. 
Fix the skeleton wire hemisphere (ii) so that the equator 
is parallel to the paper; the parallels of latitude will be 
projected as circles of true size, the meridians as radii of 
these circles. Geometrically, we draw the projection exactly 
as in geometry. Note that the scale along the circles is always 
true, but that the radial lines are foreshortened more and 
more as the distance from the centre increases. 
, 2. Stereographic. Set up the same wire hemisphere, 
with its equatorial plane vertical and parallel to the projection 
plane, and place the lamp at the further extremity of the 
diameter corresponding to the earth's axis, that is at the 
" south pole ". The parallels of latitude are again projected 
as circles, but they are enlarged, the equator being twice 
the size of the original. The meridians are projected as 
radii, as before. 

This projection has a general similarity to the orthographic, 
and its geometrical construction is a useful exercise. The 
scale is increased equally along the meridians and parallels, 
and some good Sixth Form problems may be based on the 
projection. In particular, the projection provides a ready means 
of studying the sum of the angles of a spherical triangle. 

3. Gnomonic. For this projection, the light is placed at 
the centre of the sphere. Although the parallels of latitude are 



MAP PROJECTION 



523 




o 

o 

o 

z 
o 



O 
u 
a: 
u 

fe 



o 

x 
a 



o 

i 

oc 
O 



524 



CRAFTSMANSHIP IN MATHEMATICS 



still projected as true circles, they are still more enlarged, 
and the equator itself, having the light in its own plane, 
cannot be projected at all. The general appearance of the 
projection is similar to that of the other two, but obviously 
the areas very far from the pole are greatly distorted in the 
projection. The figure shows the polar region to within 
30 of the equator. It makes a fairly good map for areas 
within 30 of the pole. 

The three projections may be usefully compared in this 
way: 

N M P Q 




Let NESQ' be a meridian of the earth, and XY its pro- 
jection plane. Take a point L at polar distance a. Then 
angle LSN |a. Let radius be R. 

Orthographic projection of arc NL = NM = R sin a. 
Stereographic = NP = 2R tan^a. 

Gnomonic = NQ = R tana. 



Observe that in the orthographic projection the outer 
circles are crowded together, in the Stereographic the outer 
are farther apart than the inner, and in the gnomonic the 
outer circles get so far apart as to be useless. It is some- 
times convenient to arrange these circles at equal dis- 



MAP PROJECTION 



525 



tances apart, and then we have the zenithal equidistant pro- 
jection. It is also possible for the distances of the parallels 
of latitude so to be regulated that the area enclosed by any 
parallel is equal to the area of the globe cut off by the same 
parallel, and then we have the zenithal equal-area projection. 
Strictly, these are not true projections, but the associated 
geometry is interesting and instructive. 



GO; 



(2) Globular Projection 

All three zenithal projections are sometimes called " per- 
spective " projections, since they can be cast as shadows. 
But the globular projection 
cannot be cast as a shadow, 
and is therefore non-perspec- 
tive. 

The geometry is a useful 
exercise for beginners. The 
projection is commonly used 
for maps of the world in two 
hemispheres. The figure repre- 
sents one hemisphere. Divide 
the equatorial diameter into 
an equal number of parts, say 
parts representing 30. Divide 
the circumference similarly. 
The curves are all arcs of circles, each to be drawn through 
three points. The mathematics of the projection is of the 
simplest. 

(3) Conical Projection 

For this we require the wire skeleton of the complete 
globe, with the light fixed at the centre. The shadow will 
be cast, not on a plane, but on the inner surface of a white 
paper cone. 

Fold up, in the usual way, a common filter paper, and 




526 



CRAFTSMANSHIP IN MATHEMATICS 



fit it into a funnel. It makes a cone with a 60 apex. A half 
circle of paper would make the same cone, the two halves 
of the diameter being brought together. A sector having 
an apex of less than 180 folds up into a more pointed cone; 
one with an apex of more than 180 folds up into a flatter 
cone. A sector of 360 (a complete circle) necessarily remains 
a plane. 

Make a white paper cone (of about 130 apex in the flat), 
slip it over the polar region of the skeleton globe so that the 
apex is in a line with the axis of the globe. The cone touches 
the sphere tangentially, viz. in a circle, and this circle is a 
parallel of latitude. If this corresponds with one of the wire 




Fig. 279 

circles so much the better. Now gently mark in the out- 
lines of the cast shadow. This is pretty easy in the neigh- 
bourhood of the line just mentioned, but the cast shadow 
gets very faint as we get farther away from the line. Now open 
out the cone on the flat (fig. 279, ii), and we have an ordinary 
conical projection. The arc represented by a heavy line 
RSR' is the circle of contact RS in (i), the " standard parallel ", 
and it is divided exactly as the circle it touches on the sphere 
is divided. 

The solid angle N of the sphere = 4 right angles. The 
angle of the cone when developed is angle RP'R'. The 
ratio of the latter angle to the former is called the constant 



MAP PROJECTION 



527 




Fig. 280 



of the cone. It is a simple Fifth Form problem to prove that 
this constant is the sine of the latitude of the standard 
parallel. 

The geometrical construction is simple. Observe that 
the parallels are arcs of circles, and that the meridians are 
straight lines. Since meridians are great circles and their 
planes pass through the centre 
of the globe, these planes must 
bisect the cone and therefore 
cut its surface in straight lines. 
The projection is commonly 
used for countries in middle 
latitudes if the latitude is not 
of too great an extent, e.g. for 
England. The conical projec- 
tion with two standard parallels 
(fig. 280) is a common projection 
for the larger European coun- 
tries. Its principle is equally 
simple. 

(4) Cylindrical Projection 

Take a large sheet of white paper and convert it into a 
cylinder of the same diameter as the skeleton wire sphere. 
Its length should be 3 or 4 times the diameter. Slip it over 
the sphere so that the equator is in about mid-position, 
and place the light at the centre of the sphere. A shadow 
of a part of the wire sphere is cast on the cylinder. Obviously 
the shadows of the two poles cannot be cast on the cylinder 
at all, and high latitudes are cast at great distances, with 
consequently great distortion. The small circle of latitude 
AB will appear as A'B'; in fact all circles of latitude will 
be projected as circles on the cylinder and will all be of the 
same size as the equator. All meridians, being great circles, 
will be cast as straight lines. Open out the cylinder on the 
flat (ii), and the projection is seen to consist of a net of rect- 
angles. E'Q' = TrEQ and may be subdivided in the usual way 



528 



CRAFTSMANSHIP IN MATHEMATICS 



The projection is not of much practical value. Except 
in the immediate neighbourhood of the equator there is far 
too much distortion. 




Fig. 281 

But various modifications of this primary cylindrical 
projection have been adopted, two of them being noteworthy: 
(1) Lambert's equal-area projection, and (2) Mercator's 
projection. 

1. Lambert's projection. Construction: divide the quadrant 




Fig. 282 



NQ (fig. 282) into an equal number of parts, say 6 of 15 
each, and draw parallels to EQ, and so obtain parallels of 
latitude. For meridians, make E'Q' TrEQ, and divide 
up into intervals of, say, 30. Note that the parallels of lati- 



MAP PROJECTION 529 

tude are horizontal lines at a distance of r sin A from the 
equator (A = lat.). 

It is a well-known theorem in geometry that the area 
between any two parallels on the enveloping cylinder is 
equal to that of the corresponding zone on the globe. Hence 
the area of the rectangle MM' is equal to the area of the globe. 
The proof of the theorem should be given. 

2. Mercator's orthomorphic projection. This is the best- 
known of all projections; it is used for navigation purposes, 
and for maps of the world. But it is responsible for many 
geographical misconceptions, for instance the misleading 
appearance of the polar areas, which are greatly exaggerated. 
Greenland is made to appear larger than South America, 
though only one-tenth its size. 

As with all cylindrical projections, the meridians are 
equidistant parallel lines; the parallels of latitude, on the 
other hand, increase in distance from one another the farther 
they are from the equator. This spacing of the parallels 
of latitudes is so arranged that at any point of intersection 
of parallels and meridians (in practice, any small area), the 
scale in all directions is the same. Hence the projection is 
orthomorphic. Literally the term means " preserving the 
correct shape ". 

The essential characteristic of the projection, then, is this 
that at any point the scale along meridian and parallel is 
the same. We give Dr. W. Garnett's ingenious illustration 
of the method of effecting this. 

Dr. Garnett takes a very narrow gore, i.e. a strip between 
two meridians on the globe (cf. the surface of a natural 
division of an orange, selected for its narrowness), and 
spreads it out as flat as possible; if very narrow there is no 
great difficulty in spreading it out very nearly flat, without 
much distortion; then it is very nearly an equal-area strip, 
i.e. its area on the flat is very nearly the same as when it was 
part of the curved surface of the sphere. The length of the 
spread-out gore is, of course, half the circumference of the 
orange. 

(E291 35 



530 



CRAFTSMANSHIP IN MATHEMATICS 



Let NAB represent the half gore, AB representing 10 
at the equator; and let NM be the central meridian. 
Divide NM into 9 equal parts, and through the points of 
division draw the parallels shown in the figure; these re- 
present 10 intervals of latitude from the equator to the 
pole. Suppose the gore to be made of malleable metal. 




30' 40' 60 



Fig. 283 * 



Hammer it out in such a way as to cause it to spread to the 
uniform width AB. Clearly we cannot do this in the im- 
mediate neighbourhood of N: there would not be enough 
metal. Hence cut the gore off at about 85. But the gore 
cannot be hammered out without expanding in length as 
well as in breadth. At, say, 40 little hammering will be 
required, and the additional length there will be slight; 
but at, say, 70 much hammering will be required to produce 




Fig. 284 

the necessary additional width, and therefore there will be 
much additional length produced. At 45 the ratio of the 
increased width to the original width is \/2 : 1, and therefore 
the length of the strip at 45 is increased \/2 times, and hence 
the area is increased there <\/2 X 1/2 times, that is, twice, 
and the thickness is therefore halved. At 60 there will be a 
doubling of width and therefore a doubling of length, i.e. 

* Figs. 283 and 284 are made to lie down, to save space. Normally, the gores 
would be given an upright position. 



MAP PROJECTION 531 

the area will be multiplied by 4 and the thickness reduced 
toj. 

It is easy to imagine the whole series of 36 gores (fig. 284 
shows 4) placed side by side, and rolled out until the edges 
meet and 36 rectangles are formed. 

Generally, every little strip parallel to the equator is 
increased both in length and breadth in proportion as the 
radius of the sphere is to the radius of the circle of latitude 
where the strip is situated. At 80 the area is increased 
about 33 times, and at 85 about 132 times. The figure 
(fig. 283) shows roughly how the gore between and 80 
is hammered out into the rectangle ABPQ. 

If the 36 gores were extended to lat. 80 N. and S., and 
placed side by side, we should have a rectangle 36 times 
AB in length and twice AP in height, and we should have the 
framework for a Mcrcator map of the world between the 
parallels 80 N. and 80 S. 

The point about the whole projection is the retention of 
true shape, though this applies to only very small areas. 
At the equator, areas are unchanged; at 80 they are in- 
creased 33 times. 

The shapes of small areas are magnified, not distorted. 
Strictly the orthomorphism is applicable only to points and 
is therefore only theoretical. 

Construction of a M creator map. The radius of a parallel 
of latitude on a sphere of radius r is r cos#. Hence if a degree 
of longitude in latitude is to be made equal to a degree 
at the equator, its length must be divided by cos0. If the 
scale of the map is to be increased in all directions in the 
same ratio, then the length of the degree of latitude measured 
along the meridian must also be increased in the same ratio. 
If y be the distance of the parallel of latitude from the 
equator in the Mercator map of a sphere of radius r, 

e 



a formula which may be evaluated by Sixth Form boys. 



532 CRAFTSMANSHIP IN MATHEMATICS 

The distances of the parallels from the equator are, in 
terms of the radius, approximately, for 



10 


176 R 


50 


1-011 R 


20 


356 R 


60 


1-317R 


30 


55 R 


70 


1-736 R 


40 


763 R 


80 


2-436 R 



These values should be checked from a Mercator in a good 
atlas: equator = 27T.R. 

Mercator, and Great Circle Sailing. The special merit 
of Mercator's projection lies in the fact that any given uniform 
compass course is represented by a straight line. All meridians 
are exactly north and south, and all parallels exactly east 
and west. Hence a navigator has only to draw a straight 
line between his two ports, and the angle this line makes 
with the meridian on the map gives his true course for the 
whole voyage. 

Any straight line drawn in any direction on a Mercator 
is called a rhumb line\ it crosses all parallels at a constant 
angle, and all meridians similarly. A sailor who is told to 
sail on a constant bearing simply sets his compass according 
to the rhumb line. 

But this course may not be the shortest; it cannot be, 
unless it is along the equator or along a meridian, i.e. along 
a great circle. A rhumb course in any other direction is not 
along a great circle, and we know that the shortest distance 
between two points in a sphere is along the great circle 
passing through them. Economy makes the navigator take 
the shortest course if he can. How is he to find it? 

A rough and ready way would be to take a wire hoop 
that would exactly fit round the equator or round one of the 
meridians (and therefore round a great circle: we neglect 
the ellipticity of the earth), hold it over the globe so that it 
passed through the two ports at the ends of the course under 
consideration, chalk in the curve, and then transfer the 
curve to the Mercator, freehand, as accurately as the corre- 
sponding graticules would allow. 



MAP PROJECTION 



533 



A navigator always follows a great circle if he can, not 
the rhumb line, and for his special use great circle courses are 
calculated and laid down on a Mercator's chart. 

If ARE is the rhumb line between two places A and B 
(the figure is a fragment of a Mercator chart), and AGCB is 
the great circle (and therefore shorter than the rhumb line), 
a navigator might sail along a series of chords AG, GC, CB, 
altering his course at G and C. He would not quite follow 




Fig. 285 



the great circle, but he would follow a much shorter route 
than the rhumb line course. 

Give the boys examples of the course between, say, 
Japan and Cape Horn, Plymouth and New Orleans, Cape 
Town and Adelaide. Let them mark in roughly both the 
rhumb line and the great circle courses on a Mercator chart. 
Remind them of the deceptive geometry, as in fig. 285, where 
the chord represents a longer distance than the arc it subtends. 

To trace the course of a great circle on a Mercator chart. 
Any great circle must cut the equator at two places and at 
a given angle. Hence it will cut (i) a given meridian at a 
point whose latitude can be determined, and (ii) a given 
parallel of latitude at a point whose longitude can be de- 
termined. 

Assume that we are given: 

(i) a, the inclination of the great circle to the plane of 

the equator; 
(ii) A, the longitude, measured from one of the points 

of section, of a meridian in latitude L. 



534 CRAFTSMANSHIP IN MATHEMATICS 

Then the following equation may be established: 

tanL = tana . sin A, 
or, sin A cot a .tanL. 

From this equation, either the latitude can be determined at 
which the great circle cuts any meridian, or the longitude 
at which it cuts any parallel. The equation may therefore 
be used to trace the course of a great circle on a Mercator 
chart. Sixth Form boys should work through a few of the 
exercises in Nunn, Exercises, Vol. II. 

Aviators are naturally much interested in great circle 
sailing. Let the boys determine an aviator's route between 
two given places, say 5000 miles apart, by stretching a string 
over a geographical globe. Then ask them how an aviator 
would set his compass. Let them lay down the course on 
a Mercator chart (graphically and approximately will do), 
and see how it differs from the rhumb line, and how compass 
directions might be determined by a succession of chords. 

(5) Sinusoidal Equal -area Projection 

This is sometimes called the Sanson Flamsteed projection; 
it is used mainly for world maps. An equal-area or " homo- 
lographic " projection is a projection where shape is sacrificed 
to equality of area. 

It differs widely from the geometrical and (mainly) 
shadow projections already considered. 

The equator (= 2?rR) is true to scale. The central meridian 
( TrR) is also true to scale. Parallels of latitude are equidistant 
horizontal lines. All the meridians are of the form of sine 
curves. Each parallel is equally divided by the meridians, 
which are nearer and nearer together towards the poles. 

Fig. 286 shows a quarter of the complete projection of the 
world map; EZ = 2NZ. Divide NZ into, say, 6 equal parts 
(of 15 each), and EZ into 6 parts (of 30 each). Each hori- 
zontal straight line is equal in length to the corresponding 



MAP PROJECTION 



535 



circle of latitude. Through the extremities of these lines 
draw the curve EN which represents the boundary of the 
quarter map. Divide every parallel into 6 equal parts, similar 
to EZ, and draw curves through the corresponding points 

N 



60 



30" 




71 



180 



60' 



Fig. 286 



of division; these curves are meridians (in the figure, half- 
meridians). 

The disadvantage of the projection is that towards the 
edge the meridians are very oblique and thus the shape is 
much distorted. The graticules along the equator and central 
meridian, on the other hand, practically retain their original 
shape. 

In any projection graticule, the horizontal lengths are 
exactly the same as in the graticule on the globe; and the 
vertical height of the projection graticule is equal to the 
length of the corresponding piece of meridian on the globe. 
Hence the area of any projection graticule is equal to the 
area of the corresponding graticule on the globe, or the whole 
area of the map is equal to the whole area of the globe. 

Each curved meridian is a sine curve: why? Might the 
sine curves be drawn before the parallels? 

The projection is very good for maps of Africa and South 
America. Why? 



536 



CRAFTSMANSHIP IN MATHEMATICS 



(6) Mollweide's Elliptical Projection 

This projection is also used for world maps. Again the 
parallels are horizontal lines. The meridians are ellipses 
(there are two special cases: the central meridian is a straight 
line and the 90 meridian is a circle). 

Again the area of the map is equal to the area of the surface 
of the globe. 

Since the area of the surface of the sphere is equal to 
4 times the area of its great circle, the area of the hemisphere 
is equal to twice the area of its circular base. 




Fig. 287 

Let the radius of the globe, fig. 287 (i), be R. Draw a circle 
(ii) of radius \/2.R (= CB). Area 2?rR 2 sq. in., which is the 
area of the half globe. Let C be the centre of the circle; 
draw a horizontal diameter ACB and a vertical diameter 
NCS. Produce AB so that CE = 2CA and CQ = 2CB. 
Draw an ellipse having EQ and NS for axes. It is one of the 
properties of the ellipse that if any line KLMN' be drawn 
parallel to EQ cutting the ellipse and the circle, KN' = 2LM; 
and as this is true for any such line, it follows that the area 
of the ellipse = twice the area of the circle = the area of 
the globe. 

Divide EQ into equal parts and through the points of 
division draw ellipses with NS as a common axis; these are 



MAP PROJECTION 537 

the meridians. Evidently all gores (e.g. NnSC, NwSw) are 
equal in area. For an equal-area projection, it remains to 
divide these gores by parallels of latitude into the same areas 
as the corresponding gores between the meridians on the globe 
are divided. This is the only difficult part of the problem. 

We have to draw KN' so that it will correspond to some 
particular degree of latitude <f> on the globe. Fig. 287 (i) repre- 
sents a section of the globe through the great circle NA'SB'. 
In fig. (ii), the circle represents the area of the hemisphere 
and the ellipse the area of the whole sphere. 

The area of the zone L'A'B'M' on the spherical surface 
(radius = R) in fig. (i) is equal to twice the area of the zone 
LABM on the plane surface (radius ^/2.R) in fig. (ii). 

We have to find the angle MCB. Let it equal a. Then 

2a -f- sin2a = TT sin9. 

It is not easy from this equation to obtain a in terms of <, 
but it is quite easy to determine cf> in terms of a. Hence if 
any parallel be drawn in the ellipse, and the angle a is measured, 
the latitude </> to which it corresponds is found at once. 
The formula should be established by the Sixth Form. 

Choice of Projection 

Let the boys examine a good modern atlas in which the 
projections used are named; and get the boys to discover 
why a particular projection is used in each case. This may 
give rise to an interesting discussion. 

Books to consult: 

1. A Little Book on Map Projection, Garnett. 

2. The Study of Map Projection, Steers. 

3. Map Projections, Hinks. 



538 CRAFTSMANSHIP IN MATHEMATICS 

CHAPTER XXXVI 

Statistics 

The Importance of the Subject 

It is highly desirable that an elementary study of this 
subject shall be included in any Sixth Form course. Statistics 
enter largely into modern science and administrative practice, 
and the underlying principles have now been so well worked 
out and have become so definite, that no large office, govern- 
ment or local , can afford to be without at least one well- 
trained statistician. The newer developments of psychology 
depend almost entirely upon a rational interpretation of 
statistics. There are some teachers who are still ignorant 
of the principles underlying the correct handling of the 
statistics of everyday school practice; and thus they are 
necessarily unable to make the most effective use of, for 
instance, an ordinary sheet of tabulated examination results. 

I have seen the subject taken up seriously in only two 
or three schools, and have therefore had little experience of 
the methods of teaching it. The teaching suggestions in 
Professor Nunn's Algebra are recognized as the most practi- 
cable yet made, and the topics he selects for inclusion in a 
school course seem to be just about right. The technical 
side of the subject is, of course, rather difficult for boys, 
but the fundamentals are easy to grasp, and it is possible 
to map out an excellent preliminary course that will give a 
good general insight into the subject and into its methods. 

The main problems to be considered may be grouped 
under the three usual heads: (1) frequency distribution of 
a series of measurements or other statistics; (2) frequency 
calculation: probability; (3) correlation. 



STATISTICS 539 

Frequency Distribution 

Frequency distribution is concerned with the best ways 
of recording statistics and of expressing most simply and 
effectively the information which they contain. Suppose, 
for instance, a Local Education Authority has examined 
20,000 children between the ages of 10 years 6 months and 
11 years 6 months for scholarships to be held in the local 
Secondary Schools. What would be the best way of recording 
the results, so that not only might their significance and its 
implications be readily apprehended but also that the record 
might form a simple means of comparison with similar 
records elsewhere? 

Suppose that all the examination papers were arranged in 
20 piles, according to the percentage of marks awarded to 
each paper, 0% to 5%, 5% to 10%, and so on up to 95% to 
100%. The height of the piles would exhibit to the eye the 
frequency distribution of the marks, the number of papers in a 
pile giving the frequency of the particular mark in that pile. 
A logically set-out record of the whole of the results might 
be called a frequency table. A column graph showing the 
number of papers in each pile would afford a useful alternative 
means of exhibiting the frequency distribution; such a fre- 
quency diagram is commonly called a histogram. 

The first thing to do is to familiarize the pupils with the 
main forms of frequency distribution in tables and diagrams. 
The records in Government " Blue Books " are often useful 
in this connexion. 

A frequency diagram differs from the ordinary, graph of 
algebra and physics. The latter represents the relation between 
two variables, or the values of a function which corre- 
spond to different values of a single variable. But a frequency 
diagram serves simply to show how often each value of the 
variable is met with in some record. 

Examples of the forms of frequency distribution may be 
drawn from very different sources from anthropometric 
measurements, from economics, from meteorology, from 



540 



CRAFTSMANSHIP IN MATHEMATICS 



medical records, and from the records of the workings of 
what (in our ignorance) we call chance. The various forms 
display resemblances that are often most pronounced when 
the diversity of origin would seem to be greatest. The re- 
semblances are brought out very clearly by the frequency curves. 
The pupils should learn that the smooth curve really 
represents the interpretation of the ideal distribution to which 
actual samples might be expected to approach if they contained 
a sufficient number of cases drawn from a field sufficiently 
wide to be really representative. When the curve is drawn 
with this idea in view, it is always of the same general pattern, 





\ 




(0 



Fig. 28 



more or less bell-shaped. The curves are, however, easily 
sorted out into seven distinct types, two of them symmetrical 
and five of them asymmetrical or skew. 

Fig. 288 (i) is an example of an almost perfect symmetrical 
curve, imposed on its histogram, which, however, is itself 
unlikely to be quite symmetrical; it is the " normal " type of 
curve. Fig. 288 (ii) is an example of a skew curve also imposed 
on its histogram. If a curve evaluated from an ordinary 
examination mark-sheet was very skew, i.e. varied con- 
siderably from the normal, it would suggest that an in- 
quiry was necessary. 

The normal curve was formerly spoken of as the graphic 
representation of the " law of error ", it being thought, 
perhaps naturally, that the mean of the distribution was the 
number (physical measurements in human beings, for in- 
stance) which represented nature's intention, deviations there- 



STATISTICS 541 

from being " errors ". But it is now recognized that ordinary 
distributions are not, even ideally, normal, and that skew- 
ness in them is almost inevitable, though probably in most 
cases the skewness or asymmetry is moderate. 

Enough should be done to teach the boys the fundamental 
fact that although when the data are few the columns which 
represent the frequencies of occurrence may exhibit no 
orderly arrangement, yet order invariably appears as the 
data become sufficiently numerous. 

There will seldom be time for pupils to deal with the 
formulae representing statistical graphs, though such work is 
certainly interesting and valuable. But the pupils should 
be made thoroughly familiar with the significance of the 
ordinary statistical phraseology. The mean is the ordinary 
arithmetical mean and is easily found when all the in- 
dividual measurements are given. The median is the middle 
measurement; it is the measurement corresponding to the 
ordinate which bisects the area of the curve; it may be 
found by calculation based upon the known properties of 
the curve, or (roughly) by using squared paper and counting 
up the squares under the curve. If we examine 5 boys and 
their percentage marks are, respectively, 85, 80, 60, 55, and 
30, the mean (average) mark is 62, and the median (middle) 
mark is 60. The lower and upper quartiles are the two measure- 
ments one-fourth from the beginning and one-fourth from 
the end of the whole series. The interquartile range includes 
the middle half of the whole series. The mode is the measure- 
ment corresponding to the highest ordinate of the frequency 
curve drawn over the histogram. 

Deviation. A boy may obtain high marks or low marks; 
a man may be tall or short; in both cases there is deviation 
from a standard. In fact, in any series there is bound to be 
" dispersion ". To measure the deviation, what standard 
should be taken? 

One recognized way of indicating the dispersion of a set 
of measurements, when something more concise than a 
frequency table is required, is to state the interquartile range, 



542 CRAFTSMANSHIP IN MATHEMATICS 

or, more usually, the semi-interquartile range, of the statistics; 
only the middle half of the measured cases is considered. This 
semi-interquartile range is often called the quartile deviation. 

Or we may strike an average of the arithmetical differences 
between the various measurements and some selected standard 
measurement, e.g. the median, the mean, or the mode. Since 
the sum of the differences of the measurements from the 
median is less than from any other standard ordinate, the 
" mean deviation " of a set of measurements is, as a rule, 
calculated with reference to their median. 

But the most useful measure of dispersion is that which 
takes account not of the deviations themselves but of their 
squares. Squaring the deviations gives greater weight to 
the larger ones, and has the mathematical advantage of making 
the numbers positive. Take the deviations, square them, 
find the mean of the squares, and then take the root of the 
mean: this gives the useful measure of dispersion called 
the standard deviation. 

Whether the subject is taught or not, the mathematical 
staff would find it a great advantage to graph their periodical 
examination results statistically. What does the graph teach? 
What does its skewness teach? What can be learnt from the 
interquartile range? What can be learnt from a comparison 
of the quartile, mean, and standard deviations? Which is 
the more useful mark in a mark-sheet, the mean or the 
median? Why? And so on. 

The fact should be impressed upon learners that the 
graphic method of presenting statistical data has advantages 
over the tabular statement of the same data, though it is 
necessary to remember that the facts cannot be more 
accurately represented by a diagram than by the data from 
which the diagram is constructed. Indeed, the diagram may, 
from imperfect draftsmanship, fall short in accuracy of the 
statistical tables it represents. The graphical presentation 
has, however, the important advantage that it presents 
lengthy series of data in a form in which the majority of users 



STATISTICS 543 

of them find it easier to grasp their sequence and their re- 
lations than when presented in the form of tables. 

Frequency determined by Calculation 

The previous section dealt with the analysis of frequencies 
actually given. It is now necessary to refer to the possibility of 
predicting them among events that have never been observed. It 
is in connexion with this problem that the topics, (1) combi- 
nations and permutations, and (2) probability, are best treated. 

The calculation of probabilities is nothing more than the 
calculation of frequencies. Probability is not an attribute 
of any particular event happening on any particular occasion. 
It can only be predicted of an event happening, or conceived 
as happening, on a very large number of " occasions ", or 
of an event " on the average ", or " in the long run ". Unless 
an event can happen, or be conceived to happen, a great 
many times, there is no sense in speaking of its probability. 
Frequency would be a better word than probability in the 
study of the subject generally, but the latter word has become 
definitely established. 

Make sure that the boys understand the notion of " in- 
dependent events ". The fall of a tossed coin is an independent 
event. Whether it will fall " head " or " tail " the next time 
it is thrown depends not at all on how it fell last time, or 
the last thousand times. If, for example, there had been 
a run of a hundred heads, the " chance " that the next 
throw would also be a head is just as great as before. 

Another idea the pupils must grasp is that frequency 
predictions are possible only in so far as the events predicted 
can be regarded as compounded of independent elementary 
events whose characteristic behaviour is already known. 
Thus, knowing that the spin of a coin is an independent 
event which will, in the long run, turn out heads and tails 
with equal frequency, we can predict with confidence what 
will happen (again in the long run) in the case of an event 
which consists in the tossing of (say) 10 coins. 



544 CRAFTSMANSHIP IN MATHEMATICS 

1. Combinations and Permutations. Apart from the 
" tricky " problems that occur in some of the textbooks 
(they are of no importance and may be ignored) this subject 
seems to be taught well. Most mathematical teachers seem 
to have neat little devices for working out nPr y nCr, &c., 
from first principles, it may be by ringing the changes syste- 
matically, and neatly classifying the results, for a group of a 
few letters on the blackboard. Even boys of average ability 
soon get to like the little stock-problems about people sitting 
round a table, or about the selection of elevens. Do not spend 
much time on the subject; it is not worth while. But give 
plenty of oral practice in such exercises as finding the value 
of 10 C 3 , 10 C 7 , &c., and do not forget the evaluation of co- 
efficients in expansions. 

2. Probability. This is a more serious topic, though its 
more elementary considerations are easily within the range 
of school work. 

Justification for teaching the subject is hardly necessary. 
It is by far the best application of the theory of permutations 
and combinations, but much more than that, it enters into 
the regulation of some of the most practical concerns of 
modern life, for instance in the use of mortality tables, 
insurance and annuity problems, and so forth. The following 
arguments and examples * may serve as a suitable introduc- 
tion to the subject. 

When we say that the probability that an event will 
happen in a certain way is I/;/, what we mean is that the 
relative amounts of knowledge and ignorance we possess as to 
the conditions of the event justify the amount of expectation. 
The event itself will happen in some one definite way, exactly 
determined by causation; the probability does not determine 
that, but only our subjective expectation of it. It is from 
this combination of knowledge and ignorance that the cal- 
culation of probability starts. 

Fundamentally, the theory of probability consists in 

* Scientific Method, pp. 260 seq. 



STATISTICS 545 

putting similar cases on an equality, and distributing equally 
among them whatever knowledge we possess. Throw a 
penny into the air, and consider what we know in regard 
to its way of falling. We know that it will certainly fall upon 
a side, so that either head or tail will be uppermost; but 
as to whether it will be head or tail, our knowledge is equally 
divided. Whatever we know concerning head, we know also 
concerning tail, so that we have no reason for expecting 
one more than the other. The least predominance of belief 
to either side would be irrational; it would consist in treat- 
ing unequally things of which our knowledge is equal. We 
must treat equals equally. 

The theory does not require that we should first ascertain 
by experiment the equal facility of the events we are con- 
sidering. The more completely we could ascertain and 
measure the causes in operation, the more would the events 
be removed from the sphere of probability. The theory 
comes into play where ignorance begins, and the knowledge 
we possess requires to be distributed over many cases. Nor 
does the theory show that the coin will fall as often on the 
one side as the other. It is almost impossible that this should 
happen, because some inequality in the form of the coin, 
or some uniform manner in throwing it up, is almost sure 
to occasion a slight preponderance in one direction. But 
as we do not previously know in which way a preponderance 
will exist, we have no reason for expecting head more than 
tail. 

Suppose that, of certain events, we know that some one 
will certainly happen, and that nothing in the constitution 
of things determines one rather than another; in that case, 
each will recur, in the long run, with a frequency in the 
proportion of one to the whole. Every second throw of a 
coin, for example, will, in the long run, give heads. Every 
sixth throw of a die will, in the long run, give ace. 

The method which we employ in the theory consists in 
calculating the number of all the cases or events concerning 
which our knowledge is equal. 

(E291) 36 



546 CRAFTSMANSHIP IN MATHEMATICS 

Let us suppose that an event may happen in three ways 
and fail in two ways, and that all these ways are equally 
likely to occur. Clearly, in the long run, the event must 
happen three times and fail two times out of every five cases. 
The probability of its happening is therefore j?, and of its 
failing, f . Thus the probability of an event is the ratio of 
the number of times in which the event occurs, in the long 
run, to the sum of the number of times in which the events 
of that description occur and in which they fail to occur. 

An event must either happen or fail. Hence the sum of the 
probabilities of its happening or failing is certainty. We 
therefore represent certainty by unity. 

The usual algebraic definition of probability is as follows. 
If an event may happen in a ways and fail in b ways, and all 
these ways are equally likely to occur, the probability of its 






happening is - -, and the probability of its failing is 
j a + b 

-- , (In mathematical works, the word " chance " is 
a + b 

often used as synonymous with probability.) 

It should be noticed that --- + -- - 1; also that 
, a -f- b a + b 

1 - = -- . Thus, if p be the probability of the 
a -}- b a + b 

happening of an event, the probability of its not happening 
is 1 p. 

When the probability of the happening of an event is 
to the probability of its failure as a is to i, the odds are said 
to be a to b for the event, or b to a against it, according as 
a is greater or less than 6. 

Suppose that 2 white, 3 black, and 4 red balls are thrown 
promiscuously into a bag, and a person draws out one of 
them, the probability that this will be a white ball is |, a 
black ball, i], and a red ball, <J. 

A few simple problems will help to illustrate the prin- 
ciples involved. 

1. What is the probability of throwing 2 with an ordinary 



STATISTICS 547 

die? Any one face is as likely to be exposed as any other 
face; there are therefore one favourable and five unfavourable 
cases, all equally likely. The required probability is there- 
fore . 

2. What is the probability of throwing a number greater 
than two with an ordinary die? Obviously there are 4 possible 
favourable cases out of a total of 6. The probability is there- 
fore or |. 

3. A bag contains 5 white, 7 black, and 4 red balls. What 
is the probability that 3 balls drawn at random are all white? 
We have 16 balls altogether. The total number of ways in 
which 3 balls can be drawn is therefore 16 C 3 , and the total 
number of ways in which 3 white balls can be drawn is 
5 C 3 . Therefore, by definition, the probability is 5 C 3 / 16 C 3 , 
that is, 5 3 . 

By a compound event , we mean an event which may be 
decomposed into two or more simpler events. Thus, fhe 
firing of a gun may be decomposed into pulling the trigger, 
the fall of the hammer, the explosion of the cartridge, &c. 
In this example, the simple events are not independent, 
because, if the trigger is pulled, the other events will, under 
proper conditions, necessarily follow, and their probabilities 
are therefore the same as that of the first event. Events 
are independent when the happening of the one does not 
render the other either more or less probable than before. 
Thus the death of a person is neither more nor less probable 
because the planet Mars happens to be visible. When the 
component events are independent, a simple rule can be 
given for calculating the probability of the compound event, 
thus: Multiply together the fractions expressing the proba- 
bilities of the independent component events. 

If, for instance, A occur once in 6 times, its probability 
is ^, or 1 for and 5 against; if B occur once in 10 times, its 
probability is y^, or 1 for and 9 against. The probability, 
or relative frequency in the long run, of the concurrence of 
the two is 0- ( y, that is, 1 for and 59 against. 

The justification of the rule may be shown thus. If 



548 CRAFTSMANSHIP IN MATHEMATICS 

two dice are thrown, the side which the one shows upper- 
most has nothing to do with the side which the other shows 
uppermost; but each die has 6 sides, each of which may fall 
uppermost, and each of these may with equal possibility 
coincide with any one of the 6 sides of the other; there are 
thus 36 possible cases, and the probability of each single 
one of them is -$-$ ( = -J- X ^). 

We may add one or two more problems. 

1 . What is the probability of throwing an ace in the first 
only of two successive throws of a single die? Here we 
require a compound even to happen, namely, at the first 
throw the ace is to appear, at the second throw the ace is 
not to appear. The probability of the first simple event is 
, and of the second |. Hence the required probability is 

A (= * x D- 

2. A party of 23 persons take their seats at a round table. 
Show that it is 10 to 1 against two specified individuals 
sitting next to each other. The probability that a given 
person A is on one side of a given person B is - 2 - 2 ; the proba- 
bility that A is on the other side of B is also -^ ; hence, the 
probability of A being next to B is - 2 5 2 = -^. Thus the odds 
are 10 to 1 against A and B sitting together. 

3. Find the probability of throwing 8 with two dice. 
With two dice, 8 can be made up of 2 and 6, 3 and 5, 4 and 4, 
5 and 3, and 6 and 2, that is 5 ways. The total number of 
ways is 3G. The probability is therefore - 6 -, and the odds 
31 to 5 against. 

4. A pack of 52 cards consisting of 4 suits is shuffled and 
dealt out to 4 players. What is the chance that the whole 
of a particular suit falls to a particular player? 

(3w!) (n\) 1 
Chance = v ~~ = approximately, 

i.e. 1 in something less than a billion 

The Laws of Probability rest upon the fundamental 
principles of reasoning, and cannot be really negatived by 
any possible experience It might happen that a person should 



STATISTICS 



549 



always throw a coin head uppermost, and appear incapable 
of getting tail by chance. The theory would not be falsified 
because it contemplates the possibility of the most extreme 
runs of luck. But the probability of the occurrence of extreme 
runs of luck is excessively slight. Whenever we make any 
extensive series of trials, as in throwing a die or coin, the 
probability is great that the results will agree pretty nearly 
with the predictions yielded by theory. Precise agreement 
must not, of course, be expected, for that, as the theory 
shows, is highly improbable. Buffon caused a child to throw 
a coin many times in succession, and he obtained 1992 tails 
and 2048 heads. The same experiment performed by a 
pupil of De Morgan's resulted in 2044 tails to 2048 heads. 
In both cases the coincidence with theory is as close as could 
be expected. Jevons himself made an extensive series of 
experiments. He took 10 coins, and made 2048 throws in 
two sets of 1024 throws each. Obviously, the probability 
of obtaining 10, 9, 8, 7, &c., heads is proportional to the 
number of combinations of 10, 9, 8, 7, &c., things chosen 
from 10 things. The results may therefore be thus con- 
veniently tabulated: 



Character of Throw. 


Theoretical 
Numbers. 


First 
Series. 


Second 
Series. 


Average. 


Divergence. 


10 Heads, Tails 


10 C = 1 


3 


1 


2 


+ 1 


9 1 


10 C! = 10 


12 


23 


17i 


+ 71 


8 2 


10 C 2 - 45 


57 


73 


65 


+ 20 


7 3 


10 C 3 = 120 


129 


123 


126 


+ 6 


6 4 


10 C 4 = 210 


181 


190 


186J 


-241 


5 5 


10 C 5 = 252 


257 


232 


2441 


- 71 


4 6 


10 C 6 = 210 


201 


197 


199 


-11 


3 7 


10 C 7 = 120 


111 


119 


115 


- 6 


2 8 


10 C 8 = 45 


52 


50 


51 


+ 6 


1 9 


10 C 9 = 10 


21 


16 


18 


+ 8 


10 


10 C 10 = 1 





1 


i 


- 1 




1024 


1024 


1024 


1024 






550 



CRAFTSMANSHIP IN MATHEMATICS 



The present writer repeated the same series of experi- 
ments, with the following results: 



Character of Throw. 


Theoretical 
Numbers. 


First 
Series. 


Second 
Series. 


Average. 


Divergence. 


10 Heads, Tails 


10 C - 1 


4 





2 


+ 1 


9 1 , 


10 d = 10 


20 


6 


13 


+ 3 


8 2 , 


10 C 2 = 45 


40 


40 


40 


5 


7 3 , 


10 C 3 = 120 


83 


150 


116i 


- 3 


6 4 , 


10 C 4 - 210 


224 


222 


223 


+ 13 


5 5 , 


10 C 5 = 252 


250 


209 


229J 


-22J 


4 6 , 


10 C = 210 


242 


222 


232 


+ 22 


3 7 , 


10 C 7 - 120 


115 


107 


111 


- 9 


2 8 , 


>C H - 45 


28 


60 


44 


- 1 


1 , 


10 C y - 10 


14 


6 


10 





10 , 


lo c iy - i 


4 


2 


3 


+ 2 




1024 


1024 


1024 


1024 






The whole number of single throws of coins amounted to 
2048 X 10, or 20,480 in all, one half of which, or 10,240, 
should theoretically give heads. The total number of heads 
obtained by Jevons was 10,352 (5130 in the first series, and 
5222 in the second). The number obtained by the present 
writer was 10,234 (5098 in the first series, and 5136 in the 
second). The coincidence with theory is in each case fairly 
close. 

Boys should be encouraged to repeat on a small scale a 
few experiments of this kind. Their interest is kindled 
when they find that a practical result closely approximates a 
theoretical estimate. 

Correlation 

Suppose that a group of measurements give us data about 
two variables, say (1) the weight, (2) the stature, of a number 
of men. Then we may not only ask questions with regard 
to the variation of weight, and questions with regard to the 



STATISTICS 551 

variation of stature, but we may also raise the further question 
of the connexion between the two. A boy who is taller than 
another is not necessarily heavier, and yet there is un- 
doubtedly some connexion between height and weight. This 
question of correlation in statistical theory is becoming one 
of rapidly increasing importance. 

The existence of the connexion itself may, of course, be 
in question. Is a boy who is good at sports likely, in the 
long run, to be a duffer in the classroom? Some very mathe- 
matical pupils are, and some are not, musical. Some very 
musical pupils are, and some are not, mathematical. Is it 
possible to discover a definite measurement of the degree 
of connexion between two things whenever the things them- 
selves are capable of trustworthy estimation? 

One of the simplest methods of measuring correlation is 
Professor C. Spearman's foot-rule method; it is easily 
mastered in five minutes. Whatever work of this kind may 
be attempted with schoolboys, not only the Spearman co- 
efficient, but the Bravais-Pearson coefficient, should be 
familiar to all teachers. 

From the teaching point of view, Spearman's method 
possesses the advantage that original material for illustrating 
its use is always available in schools. Investigations of the 
correlation between the performance of a class in different 
subjects, in the same subject in different terms, in different 
examinations in the same subject, in school performances 
which are not all academic subjects, all these would give 
valuable information to the teacher. The use of the corre- 
lation coefficient as the measure of the " reliability " of an 
examination test is of special importance. 

The teacher should consult the works of Professor Spear- 
man, Professor Thorndike, Professor Karl Pearson, Dr. W. 
Brown, Udny Yule, and A. L. Bowley. 

Statistics has become such a big subject that teachers 
may decide against its introduction into schools. But at 
the very least boys should be warned of the seriously faulty 



552 CRAFTSMANSHIP IN MATHEMATICS 

inferences drawn from statistics by the imperfectly-trained 
student of economics. University degrees in Economics may 
now be obtained by students with only a superficial know- 
ledge of mathematics. Need we therefore feel surprised at 
the absurd economic opinions now often expressed by some 
of our younger politicians? One of the commonest political 
fallacies is to impose a correlation on two utterly unrelated 
graphs, perhaps those concerning (i) foreign trade and (ii) the 
marriage rate, on the sole ground that the graphs show some- 
what similar variations. 

Mathematics teachers should warn their pupils that opinions 
based on statistics cannot be more than probably true; the 
degree of probability may be very great, but there can be no 
absolute certainty. 

Statistics is beginning to occupy an important place in 
theoretical physics. Dirac says: " When an observation is 
made on any atomic system ... in a given state, the result 
will not in general be determinate, i.e. if the experiment is 
repeated several times under identical conditions, several 
different results may be obtained. If the experiment is re- 
peated a large number of times, it will be found that each 
particular result will be obtained a definite fraction of the 
total number of times, so that we can say there is a definite 
probability of its being obtained any time the experiment is 
performed. This probability the theory enables us to cal- 
culate. In special cases, the probability may be unity, and the 
result of the experiment is then quite determinate." Instead 
of the accuracy and precision which until a short time ago we 
have always ascribed to nature, we seem to have nothing but 
uncertainty and randomness. Nature seems to know nothing 
whatever of simple mathematics. Virtually the present-day 
physicist seems to be immersed in the study of the statistics 
of electron " jumps ". We can foretell what will happen in the 
long run when we throw up coins, and apparently we can quite 
definitely forecast what will happen in the long run when we 
experiment with vast crowds of atoms and electrons. The 
laws of averages and of probability are entering more and more 



STATISTICS 553 

into the physics of small-scale things. The 2000-year-old ques- 
tion of causation (determinism) presents itself anew. 



CHAPTER XXXVII 

Sixth Form Work 

The Normal Programme for Specialists 

The work done by Sixth Form specialists is almost always 
work in preparation either for University Scholarships or 
for the Higher Certificate. It has become stereotyped in 
scope, and much of it has been described as " deadly dull ". 
Inasmuch, however, as the University Authorities seem to 
require sent up to them boys who have been "well grounded ", 
boys who are proficient in the use of those mathematical 
weapons which will make attack on the University Course 
immediate and effective, mathematical teachers appear to 
have no option but to make their boys face the necessary 
drudgery. If boys are actually going on to the University, 
perhaps that does not much matter. But if they are not, 
it is pretty safe to say that their mathematical interest will, 
as a rule, cease as soon as they leave school. 

In 1904, a Committee of the Mathematical Association, 
consisting of 34 of the leading mathematical masters in the 
country, reported on " Advanced School Mathematics ". 
The committee took into account the different classes of 
boys who study advanced mathematics in schools, e.g. 
candidates for army examinations, science students, engineer- 
ing students, and boys who intend to read mathematics at 
the University, and they framed a course of instruction which, 
they hoped, would prove suitable for all. The following is a 
summary: 



554 CRAFTSMANSHIP IN MATHEMATICS 

1. Algebra: partial fractions, elementary manipulation with 
complex numbers and geometrical applications thereof, the 
theory of equations so far as it treats of the numerical solution 
of equations, the notation and easy properties of determinants, 
the simpler tests of convergency, and the binomial, expo- 
nential, and logarithmic series; but excluding the theory of 
numbers, probability, continued fractions, and advanced 
theorems on inequalities, on indeterminate equations, and on 
summation of series. 

2. Differential and Integral Calculus: introduction, and 
a free use of the calculus in subsequent work. 

3. Trigonometry: graphical illustrations of De Moivre's 
theorem, simple work in trigonometrical series and factors. 

4. Conic Sections: a treatment of the elementary parts 
of the subject, in which either the geometrical or the ana- 
lytical method is used, that method being used in each 
particular case which is most suitable for the problem under 
discussion. 

5. Solid Geometry: the elementary geometry of the plane, 
cone, cylinder, sphere, and regular solids, including practical 
solid geometry. 

6. Dynamics: an introduction to the dynamics of rotation 
(in two dimensions), viz. the motion of a rigid body round 
a fixed axis with uniformly accelerated angular velocity, 
together with other simple cases of the motions of rigid 
bodies. 

The Committee urged the importance of " a more intimate 
union between the teaching of mathematics and science, 
whereby theoretical and practical work may be brought into 
relation with one another ". 

In 1907, the committee outlined a special schedule of 
work suitable for boys preparing for Oxford and Cambridge 
scholarships. 

1. Pure Geometry. Geometry of straight lines, circles, 
and conies; inversion, cross-ratios, involution, homographic 
ranges, projection, reciprocation and principles of duality, 



SIXTH FORM WORK 555 

elementary solid geometry including plans and elevations of 
simple solids. 

2. Analytical Geometry. Straight lines and curves of 
the second degree; tangential co-ordinates. Excluding 
homogeneous co-ordinates, invariants, and analytical solid 
geometry. 

3. Algebra, including elementary theory of equations. 
Excluding recurring continued fractions, harder tests of 
convergence, theory of numbers, and probability. 

4. Geometrical Trigonometry. Excluding spherical trigo- 
nometry. 

5. Analytical Trigonometry. Properties of circular, hyper- 
bolic, exponential and logarithmic functions. Excluding 
the proofs of the infinite products for sine and cosine, and of 
the series of partial fractions for the other trigonometrical 
ratios. 

6. Calculus. Total and partial differentiation; Taylor's 
and Maclaurin's theorems; elementary integration; simple 
applications to plane curves (especially to such as are of 
intrinsic importance), to maxima and minima, to areas and 
volumes, and to dynamics; curve-tracing, not as a rule to scale. 
Excluding the theory (but not the use) of differential equations. 

7. Dynamics. Elementary statics, including simple gra- 
phical statics; elementary kinematics and kinetics, including 
motion of a rigid body about a fixed axis, and motion of 
cylinders and spheres in cases where the centre of gravity 
describes a straight line. Excluding hydrostatics and hydro- 
dynamics. 

In a suggestive article which appeared in the Mathematical 
Gazette for January, 1923, Mr. F. G. Hall asks for a much 
greater unification of subjects in Sixth Form work. He 
points out that the separate consideration of the different 
subjects involves a great waste of time. He says that, for 
instance, ratio and proportion are treated algebraically, 
geometrically, and trigonometrically; logarithms occur in 
every type of textbook; variation is considered from the 



556 CRAFTSMANSHIP IN MATHEMATICS 

point of view of formal algebra, algebraic graphs, trigo- 
nometrical graphs, and the calculus. He then outlines a 
revised scheme, under seven headings: 

1. Ratio in algebra, geometry, and trigonometry. 

2. The inter-relation of trigonometry and geometry. 

3. Variation; the general study of functionality. 

4. The elements of the differential calculus and its application to 
algebra, geometry, and trigonometry. 

5. Logarithms and their use in arithmetic, algebra, and trigono- 
metry. 

6. The elements of the integral calculus and its application to 
mensuration. 

7. Elementary Analysis: (a) the important expansions of algebra 
the binomial, exponential, and logarithmic theorems; (p) further 
trigonometry, with additional work on De Moivre's theorem and the 
expansions to which it leads; (y) easy treatment of the following: 
Rolle's theorem and the First Mean-Value Theorem; the different 
substitutions employed to effect the important types of integration; 
successive differentiation; elementary differential equations. The 
underlying ideas of this section are (i) the analytical study of " form " 
in pure mathematics, and (ii) the development of manipulative power 
to enable the study of Higher Mathematics to be undertaken when 
the pupil proceeds to a University course. 

The whole article is worth reading; it makes fruitful sugges- 
tions for economy of effort and for the linking up of different 
subjects. 

Professor Nunn, discussing studies of the kind usually 
found in textbooks on " higher algebra ", urges that such 
studies do not offer the most suitable course of instruction 
for the general body of Sixth Form pupils. " For the student 
who is to be a teacher or engineer, or to engage in higher 
industrial or administrative work, as well as for the student 
who is continuing his mathematical studies as part of a 
general education, the best course would seem to be one 
which sets in clear relief the central aims and most vital 
notions of the main branches of mathematics, supplements 
exposition with sufficient practical exercises to give the 



SIXTH FORM WORK 557 

student a real training and the sense of mastery that comes 
with training, and, in particular, illustrates vividly the essential 
part which mathematics plays in so many departments of 
modern life and activity. " 

Excellent advice. Would that it were more generally 
followed! 

Simpler Fare for the Non- Specialists 

It will be observed in the last paragraph that Professor 
Nunn was speaking of the " general body " of Sixth Form 
pupils, whereas both the Committee of the Mathematical 
Association and Mr. F. G. Hall had in view the small 
section who intend to specialize in mathematics. Now it 
is a fact, and a very regrettable one, that Sixth Form boys 
who are not specializing in mathematics and science, or at 
least the great majority of them, do no mathematics at all. 
It is for these boys that I wish to enter a plea. I do not ask 
for a supersession of the present type of specialist Sixth 
Form work; in practice such supersession is not possible. 
The Universities know what they want and schools have no 
option but to prepare the boys accordingly. But I do ask 
for the provision, for all sections alike of the Sixth Form, say 
for two periods a week (assuredly a modest allowance), for 
a course of mathematical instruction that shall be at once 
less bookish, less academic, more informal, more interesting, 
in short, mathematics of the by-ways rather than the sterner 
stuff so much beloved by the successful mathematician. 
I want all boys to leave school with a cultivated interest in 
mathematics. Some people are of opinion that this may be 
effected by making the proposed general Sixth Form course 
wholly recreational, but I would make the work more exact- 
ing than that. I believe it might continue to be as exacting 
as the work already done in the Fifth. Nevertheless, I would 
willingly sacrifice much of the formality of the subject in 
order to ensure a permanence of interest. The formal 
training I would willingly subordinate to interest and 



558 CRAFTSMANSHIP IN MATHEMATICS 

knowledge, especially knowledge of unsuspected points of 
contact between mathematics and nature. 

With this purpose in view I throw out a few hints in 
seven short chapters: 

1. Harmonic Motion. 

2. The Polyhedra. 

3. Mathematics in Biology. 

4. Proportion and Symmetry in Art. 

5. Numbers; their unexpected Significance. 

6. Time and the Calendar. 

7. Mathematical Recreations. 



CHAPTER XXXVIII 

Harmonic Motion 

The Ordinary Book work of S.H.M. 

The work commonly done in connexion with Simple 
Harmonic Motion (which has already been touched upon 

in Chapter XXXI) usually begins 
in this way: 

Let P travel with uniform speed 
round the circumference of a circle. 
If Q is the foot of the perpen- 
dicular on the diameter AB, the 
motion of Q is an oscillation be- 
tween A and B. Q is the projec- 
g. 289 tion of P on AB, and the motion 

of Q is Simple Harmonic Motion. 

Thus we may define S.H.M. as the projection of a uniform 
circular motion on to a diameter of the circle. 

Then usually follows a series of blackboard demonstrations: 




HARMONIC MOTION 559 

Let the amplitude of the oscillation, i.e. the radius of 
the circle, be a. 

Let ft be the square of the angular velocity of the point P 
about the centre O. 



(i) Since the angular velocity of OP = 

/. the linear velocity of P a\//x- 

The acceleration of Q /z.OQ towards O; 

acceleration of Q 

or, . = 



_ acceleration of Q 
displacement 

(ii) The period of oscillation . 



(iii) The velocity of the point Q (in any position) 

" 



(iv) The period of a small oscillation of a simple pendu- 
lum = 2?r A/ - . 
of 

(v) The length of a seconds pendulum = / %> 

This is all sound enough, but to many boys it is just an 
affair of algebra, with only the vaguest relation to practical 
life. The boys probably forget their practical exercises on 
loci in geometry when from models of linkages they learnt 
how a to-and-fro motion may be converted into a circular 
motion, or, if they remember, they do not associate the new 
work with the old. They are probably not made to realize 
that S.H.M. forms the basis of the investigation of most 
oscillatory movements that occur in nature, such as the 
small oscillation of a simple pendulum, the vibration of 
strings and other bodies emitting musical notes, the light 
vibrations in the aether, the vibrations producing Fletcher's 
trolley-wave; and so on. They have been studying S.H.M. 
in their mechanics and physics without knowing it, and 
now they are studying S.H.M. as something having only the 
vaguest relation to their mechanics and physics. 



560 CRAFTSMANSHIP IN MATHEMATICS 

What is the use of playing about with the mere algebraic 
formula of simple harmonic motion before the motion has 
been studied and its full significance grasped in the related 
practical work in mechanics and physics? The laboratory, 
not the classroom, is the place for teaching S.H.M. 

S.H.M. Experiments 

Elaborate experiments are not necessary, Here are a 
few simple ones. 

1. Let the boy stand facing a white wall, or a lantern 
screen, and at the back of the room let a strong light be 
placed at about the level of his head. His shadow will be 
cast on the wall or screen. Now let him whirl, in a horizontal 
plane, round his head a small heavy body attached to the 
end of a string about a yard long. After a little practice he 
can maintain his hand in almost the same position, and 
keep on whirling at a fairly constant velocity. As the small 
heavy body moves round his head, its shadow moves to 
and fro on the screen. There is a true projection of the 
circular motion, the shadow showing a true S.H.M. (A 
conical pendulum may also be used. Let the heavy body 
first hang vertically, then cause it to swing in such a way 
that the pendulum describes a conical surface. Note that 
the effective length of this pendulum is the vertical depth 
of the weight below the point of support.) 

2. Let the boy continue to whirl the string-pendulum 
around his head; then set up an ordinary pendulum between 
him and the wall, and let it vibrate. The latter may be allowed 
to swing with rather more than " small " vibrations; these 
will not, it is true, be in strict S.H.M., but for a rough experi- 
ment that will not much matter. Now adjust the length 
of the string in the first experiment so that the to-and-fro 
movement of its shadow is about equal in length to the 
swing of the suspended pendulum, and so adjust the time of 
whirling that there is one complete revolution to a complete 
to-and-fro movement of the suspended pendulum. With 



HARMONIC MOTION 



561 



a little practice (it needs practice), the shadow of the heavy 
weight may be made to follow exactly the movement of the 
pendulum bob (except of course that the latter does not 
move quite in a horizontal plane). The experiment convinces 
the observer that the movement of the shadow is syn- 
chronous at all points with the movements of the pendulum 
bob. 

3. Mount on a board a good sine curve as produced by a 
Fletcher's trolley running with uniform speed. Cut a slit 
in a piece of cardboard and arrange the cardboard so that 



Fig. 290 

its mid-line AB always coincides with the axis XY of the 
curve. Move the cardboard to and fro with uniform speed x. 
The bit of curve showing through the slit seems to move up 
and down with S.H.M. A permanent model is worth 
making in the school workshop. A groove may be run along 
the board, coinciding with the XY axis, and at the back of 
the cardboard (a thin piece of wood is better) two bits of 
wood are fixed at AB, one on each side of the slit, to slide 
along the groove. A better model may be devised by mount- 
ing the curve on a large wooden cylinder which is made to 
revolve, the cardboard with the groove being fixed medially 
in front of it. The resemblance of the bit of line moving 
up and down the groove to a pendulum movement is very 
striking. 

4. Revise the laboratory experiments on the pendulum, 
and verify that the time of vibration of a pendulum varies 
as the square root of the length. 

Lead the boys to see clearly that Fletcher's vibrating 
lath is really a pendulum, and the trolley simply a device 
for recording its vibrations pictorially; and that the model 

(291) 37 



562 



CRAFTSMANSHIP IN MATHEMATICS 



in (3) above is, in its turn, another device for showing the 
lath vibrations more slowly, so that we can examine the way 
in which the lath really did move. 

The term " harmonic " is justified because when the 
vibrations of the lath follow one another with sufficient 
rapidity, a definite musical note is heard which rises higher 
in pitch with increasing rapidity of the vibrations. So with 
vibrating things generally. 

Compound Harmonic Motion 

This subject is important in connexion with the study 
of sound. Lissajous curves are produced by compounding 
harmonic motions. The " compound pendulum " produces 

many of them in a simple and 
sufficiently effective manner. 
Here the mathematical master 
will be able to get help from 
the physics master. 

A pendulum dropping sand 
affords the simplest means of 
illustrating the combined mo- 
tions. A funnel F containing 
sand is suspended by two 
strings, passing through two 
small holes in a cork C, from 
two hooks A and B in the beam 
of the frame MN, and swings 
in S.H.M. at right angles to the 
frame; the sand dropped from 
the funnel makes a line on the 

paper below, also at right angles to the frame. If the 
paper is now steadily drawn across the baseboard with 
correct time adjustment, in a direction parallel to the frame, 
the sand will trace out a sine curve. Now draw the funnel 
aside so that it swings in the plane of the frame and the 
sand makes a line in that plane; the swing is again in S.H.M.; 




Fig. 291 



HARMONIC MOTION 563 

slide the paper across the board, perpendicularly to the 
plane of the frame, and another sine curve results, but at 
right angles to the former. Finally, draw the funnel aside 
in a direction neither parallel nor perpendicular to AB (it 
might have been fastened back by a piece of cotton attached 
to the upright Z, fixed in a 45 position, and suddenly 
released by cutting or firing the cotton). The new motion 
combines the characteristics of the two former S.H.M.s, 
and the motion is said to be compound H.M. The precise 
form of the movement will depend (1) on the ratio of the 
two S.H.M.s, that is, the ratio of the lengths of the two really 
separate pendulums DE and CE, D and C being the re- 
spective points of support; and (2) the direction in which 
and the distance to which the funnel is drawn back. We 
may modify the first of these factors by sliding the cork C 
up and down the string; the length of the shorter pendulum 
will always be equal to CE. Of course the paper will not 
now be moved at all. 

The pupils must understand clearly that if the pendulum 
swings at right angles to the frame MABN, its effective 
length is DE; that if it swings parallel to the frame its effective 
length is CE; that therefore by altering the position of C 
we may make the ratio of the two lengths any value we 
please. 

Let the ratio be 1:4, e.g. let CE be 9" and DE 36". 
Set in motion by releasing from Z. The bob (the funnel) 
of the two pendulums cannot move in two 
directions at the same time and it therefore 
makes a compromise and follows a path 
compounded of the two directions. It 
traces over and over again fig. 292. Since 
the times of vibration vary as the square 
roots of the lengths of the pendulum, these ^ Fig. 292 
times are as \/l : -y/4, i.e. 1 : 2. Thus the 
short pendulum CE swings twice while the larger one DE 
swings once. 

If we wish the times of vibrations to be 2 : 3, that is if 




564: 



CRAFTSMANSHIP IN MATHEMATICS 



we wish the larger pendulum to swing twice while the shorter 
swings three times), the ratio of the pendulum lengths must 
be 2 2 : 3 2 or 4 : 9, i.e. the shorter pendulum 
must be 4/9 of the larger one. If the larger 
one DE is 36", the shorter one CE must 
be 16", and the cork C must be adjusted 
accordingly. The sand curve now traced, 
over and over again, is shown in fig. 293. 

Let the pupils try other simple ratios. 
Here is a short table of musical intervals 
formed by two notes which are produced 
by numbers of vibrations bearing to each 
other the same ratios as those given in the first column. 




Fig. 293 



Frequency 
Ratio. 


Pendulum 
Length Ratio. 


Corresponding 
Musical Interval. 


1 2 
2 3 
3 4 


4 1 
9 4 
16 9 


Octave (fig. 292) 
Fifth (fig. 293) 
Fourth 


4 5 
5 6 


25 16 
36 25 


Major Third 
Minor Third 


3 5 


25 9 


Sixth 



The new and more elaborate figures will interest the pupils, 
who must, however, realize that in all cases the movement 
actually executed is the resultant of two S.H.M.s perpendicular 
to each other. 

In practice it is virtually impossible to set the two pen- 
dulums at the exact ratios given. The simple curves are 
therefore not maintained, but they open out and close again 
in a curious but regular movement. Here are examples of 
the changes seen in figs. 292 and 293, representing the octave 
and the fifth. But within a short time the curves are lost in 
the gradually spreading sand. The well-known successive 
phases of Lissajous figures, due to the tuning-forks not 
being in exact unison, are identical with the figures here 
shown, and, for all practical purposes, are produced in the 



HARMONIC MOTION 



565 













Fig. 294 

same way. The boys should see them produced in the physical 
laboratory. 

Obviously a better device than the swinging sand funnel 
is wanted for producing 
the figures. Pupils often 
suggest the substitution 
of a pencil or a pen for 
the sand, but of course 
this is not possible, 
inasmuch as the funnel 
does not move in a hori- 
zontal plane. 

This difficulty has 
been overcome by devis- 
ing an entirely new type 
of compound pendu- 
lum. Two pendulums 
are hung from a small 
wooden table supported 
on three legs, the pendu- 
lum rods passing through Figt 295 
large holes in the table- 
top so that they can swing without touching it. To the top 
of one, a small table with paper is fixed so that it moves to 




566 CRAFTSMANSHIP IN MATHEMATICS 

and fro while the pendulum swings. The top of the other 
carries a long rod which in turn carries a glass pen. Each 
pendulum can swing in one direction only, like a clock 
pendulum. Each pendulum swinging alone would simply 
describe a straight line of ink, again and again, until the 
motion died away. If the two pendulums are made to swing, 
not together, but in succession, there will be two straight 
lines at right angles to each other. But the two may be made 
to swing together, each to record whatever impulse may 
first be given to it, and so we get figures of the same type 
as the sand figures. 

As the pen moves in a vertical plane, the surface of the 
receiving table top of the other pendulum is given the neces- 
sary slight cylindrical curvature, in order that the pen may 
always be just in contact with it. 

We may vary as we please the impulses given to the 
pendulums. We may start them either at the same time, or 
one rather later than the other, say when the first has covered 
some definite fraction (J, |, or f) of its path. We may vary 
the length of either pendulum by raising or lowering the 
weights (bobs); the effective length of a pendulum is from 
the point of suspension to the centre of the bob. 

Thus the figures produced may be varied greatly. 

If the two pendulums are exactly equal, and if we start 
them swinging at exactly the same instant, we shall obtain 
one of these figures: 






Fig. 296 



These are really only different phases of the same figure, and 
they form the simplest group of Lissajous figures: all are 
ellipses, the straight line and the circle being merely par- 
ticular cases. If the two are started at the same pace and 
at the same moment, the figure is a straight line; if the one 



HARMONIC MOTION 



567 



begins its movement when the other has already completed 
half its path, the figure is a circle; if the difference is greater 
or less than half a path, then an ellipse. 

But although the pendulum continues to move, the pen 
does not continue to mark out exactly the same line, circle, 
or ellipse. We have to take friction into account. When 
the pen comes round to the point where, say, the circle 
began, it just misses it, and the second circle begins inside 
the first: really we get a spiral, ending at the centre of the 
paper when the pendulum comes to rest. By varying the 
friction, varying the ratio of the pendulums, and varying 
the impulses, we may obtain figures of extraordinary beauty. 
Many of these, exquisitely reproduced in colour, may be seen 
in Newton's Harmonic Vibrations and Vibration Figures. The 
more elaborate pendulums which have been designed for 
producing such figures (Benham's triple pendulum, for in- 
stance) are worth a careful examination for their mechanical 
ingenuity alone. 

The mechanical difficulties associated with the compound 
pendulum (the harmonograph) are many, and only the 
exceptionally patient teacher is advised to purchase one. 
The feeding of the glass pen with a suitable ink, and the 
adjusting of the pen, are particularly teasing operations. 

In two or three cases 
I have known the beauti- 
ful figures make a strong 
appeal to pupils who had 
previously professed their 
dislike of mathematics. 

Various other instru- 
ments have been designed 
for producing curved de- 
signs mechanically. Fig. 
297 shows two designs pro- 
duced by the Epicycloidal 
Cutting Frame, but, gene- 
rally speaking, the mathe- Fig. 297(0 




568 



CRAFTSMANSHIP IN MATHEMATICS 




Fig. 297 (") 

matics of such designs, though not advanced, is too tedious and 
elaborate to render its introduction into schools worth while. 




Geometrical Construction of the Compound Curves 

Let the pupils draw a few of the simpler figures geo- 
metrically. 

They may refer again to 
the pendulum whirling round 
the head. Let a revolution 
take two seconds, so that the 
pendulum rotates through y^- 
of the circle in -g- of a second. 
To an observer watching the 
shadow on the screen, the 
bob appears to travel from 
1 to 2 in the straight line, 
while it really travels from 1 
to 2 in the circle. It appears 
to travel with greatest velocity 
at 1 and 9 in the straight 
Fig . 29 8 line, and to be momentarily 



\ 



1514 15 
12 II 



16 
10 



I 
9 



45 
6 



HARMONIC MOTION 



569 



at rest at 13 and 5; also to travel from A to B in 1 second, 
though in that time it has really travelled half-way round 
the circle. A comparison of the unequal distances covered 
in equal times, in AB, serves to show the varying velocity 
of a simple pendulum. 

Remember that the phase of any point in S.H.M. is the 
fraction of a period that has elapsed since the point last 
passed through the position of maximum displacement. 




Fig. 299 

Here is a figure (fig. 299: an ellipse) produced by two 
S.H.M.s of the same period but differing in phase by \ of 
a period. Semicircles suffice for obtaining the necessary 
projectors. The points for the ellipse are determined by the 
intersection of the two sets of projectors. 

Projectors from a and 3 (^ of a phase apart) produce the 
point m. 

Projectors from b and 2 (^ of a phase apart, yg- in each 
circle) produce the point n. And so on. The curve follows 
the route marked out by the diagonals of successive parallelo- 
grams. 



570 



CRAFTSMANSHIP IN MATHEMATICS 



Boys who are poor draughtsmen sometimes make a hash 
of more complex figures, generally, however, because they have 
not grasped the main principle which is the same for all cases. 

Here is another figure (fig. 300) showing the compounded 
motions of two pendulums whose lengths are as 16 : 25 and 
whose vibration ratio is therefore 4:5. Divide the semicircle 
EEC into 8 equal parts and the semicircle ADB into 10. 




Fig. 300 

Project on to the respective diameters, and continue across 
the square so that the projectors intersect as shown. The 
phases are intended to be the same, so that the two pendulums 
may be supposed to start together at B. The pendulum swing- 
ing towards A would reach M when the pendulum swinging 
towards C reached N i.e. if the two were swinging in- 
dependently; but inasmuch as they are moving together 
their motions are compounded; neither is able to take its 
own path, and the path actually followed is BR. 

Observe that all the separate bits of line, horizontal and 
vertical, in the square, represent equal intervals of time. 



HARMONIC MOTION 571 

Hence every point of intersection through which the curve 
passes is determined by compounding two distances re- 
presenting equal intervals of time. But after the point Q, 
we have, of course, to begin to allow for a reversal of direction. 
The curve simply follows the route marked out by the diagonals 
of successive parallelograms. 

Mathematically, the subject is hardly worth following 
very far, unless the pupils are seriously studying the subject 
of sound in physics. But there are a few exercises in Nunn's 
Algebra that all Sixth Form pupils might profitably work 
through. 

Spirals 

This subject of spirals, though outside the scope of the 
chapter, may be conveniently mentioned here. 

A little work on spirals is worth doing, if only to emphasize 
the fact that there is a practical side to the notion that angles 
may have any magnitude. For instance, in Archimedes 9 
spiral, r = ad\ in the Logarithmic spiral, r = a B \ in the 
Hyperbolic spiral, r9 a- in the Lituus, r*9 a. A little 
work on roulettes should also be done cycloids, epicycloids, 
hypocycloids, but not trochoids. These are all best treated, 
for the most part, geometrically. Such peculiarities exhibited 
by curves as asymptotes, nodes, cusps, points of inflexion, 
&c., should be explained. 

All these curves are full of interest; dwell on that side 
of them. 

Spirals afford a good start for the study of vectors. In 
the Archimedean spiral, equal amounts of increase in the 
vectorial angle and radius vector accompany one another, 
i.e. if one is in A.P., so is the other. In the logarithmic spiral 
if the vectorial angles form an A.P., the corresponding radii 
form a G.P. 

Books to consult: 

1. Any standard work on Sound 

2. Harmonic Vibrations, Martin. 



572 



CRAFTSMANSHIP IN MATHEMATICS 



CHAPTER XXXIX 

The Polyhedra 

Euclid, Book XIII 

It is to be regretted that the displacement of Euclid 
has led to the abandonment of the greater part of the substance 
of his 13th book. It is true that the five regular solids are 
referred to in most modern textbooks of geometry, sketches 
of them given, " nets " for their development suggested, 
and perhaps Cauchy's proof of Euler's relation E+2 = F+V 
worked out. But assuredly rather more than this ought to 
be done. The intimate relations of the five solids to each 
other, and the almost equally close relations between them 
and their many cousins, are so remarkable, and at the same 
time are so simple, that all boys ought to know something 
about them. The work forms a fitting completion to the 
course of solid geometry. 

A regular polyhedron is one all of whose faces are equal 
and regular polygons, and all of whose vertices are exactly 
alike and lie on the circumscribed sphere. There are only 
5, and the proof that there cannot be more is easy and should 
be provided. They are: 



Name. 


Number and 
Nature of Faces. 


Sum of Plane /.s 
at Vertex. 


Remarks. 


Tetrahedron 


4 equilateral As 


60 X 3 = 180 


} Found as 


Cube 


6 squares 


90 x 3 = 270 


> natural 


Octahedron 


8 equilateral As 


60 X 4 240 


] crystals 


Dodecahedron 
Icosahedron 


12 pentagons 
20 equilateral As 


108 X 3 - 324 
60 X 5 = 300 


\ Artificial 



Closely related to the 3 natural polyhedra are 2 other 
natural solids, the rhombic dodecahedron (faces = 12 rhombuses), 
and the trapezohedron (faces = 24 kites). And closely related 



THE POLYHEDRA 



573 



to the 2 artificial polyhedra are 2 other artificial solids, the 
triacontahedron (faces 30 rhombuses), and the hexacon- 
tahedron (faces = 60 kites). In all 9 cases, E + 2 = F + V. 









Tetrahedron 




Cube 




Octahedron 




Dodecahedron 




Icosahedron 
Fig. 301 







574 



CRAFTSMANSHIP IN MATHEMATICS 



On the previous page are sketches of the 5 polyhedra and 
nets for their construction. Third Form boys make them 
up readily enough (for practical hints, see Lower and Middle 
Form Geometry, pp. 202-3). 

In my own teaching days, we taught the interrelations of 
all 9 solids, but in these days of high pressure, one has to 
be content with a few main facts about the 5 regular solids 
alone. For those who wish to construct models of the 

A other four, the 

following di- 
mensions of the 
respective rhom- 
buses and kites 
may be useful 




Fig. 302 




* J" 

Rhombic dodecahedron, - = - . 

V ' +V. A MP 2 

Inacontahedron, = . 



(fig. 3 2 ) : 



T . U A MP 

Tnacontahedron, _ 

,, AC 3V3 AE 1 

Trapezohedron, _ = ^^ = -. 

- - AC V25 + 5V5 AE 

Hexacontahedron, _ = -- ; 



The key propositions to Euclid XIII are 13-18. The 
first five of these show how to construct the polyhedra, how 
to inscribe each in a sphere, and how to express the length 
of an edge in terms of the radius of the sphere. If E be the 
edge, and R be the radius of the sphere, 

Proposition 13, tetrahedron, E = |- \/6 . R. 

14, octahedron, E = V 2 R - 

15, cube, E = fi/3 . R. _ 

16, icosahedron, E = Vl<K/5(<v/6--l) R - 

17, dodecahedron, E = ^ ( V5 - 1)R. 



THE POLYHEDRA 



575 



Encourage the pupils to seek for interrelations amongst 
these. For instance, compare Propositions 15 and 17; in 
the latter, E = |(V 5 1) of E in the former. Hence, if 
the edge of the inscribed cube be cut in golden section, the 
length of the greater segment is the length of the edge of the 
inscribed dodecahedron. 

Euclid XIII, 18, is also useful. To set out the edges of 
the five regular solids and to compare them with one another 
and with the radius of the circumscribing sphere. 

Cutting One Polyhedron from Another 

We shall refer to polyhedra " contained " within other 
polyhedra. The following volume relations should there- 
fore be noted: 





Contained Solid. 


Containing Solid. 






Octahedron. 


Cube. 


Tetrahedron. 


Octahedron 





9:2 


27 :2 


Cube 


6: 1 





3:1 


Tetrahedron 


2: 1 


9: 1 






The mutual relations of the two artificial solids are less simple. 
It may be noted that the ratio of the edges of an icosahedron 
and its contained dodecahedron is 6 : 1 -j~ -y/5. 

A group of interesting problems are those of calculating 
the angles between edges, between faces, and between edges 
and faces, in the polyhedra. They are good problems, but 
some are a little tedious. In this connexion, the so-called 
15th book of Euclid may be usefully consulted. 

Practical demonstrations of polyhedral interrelations always 
appeal to boys. Encourage the boys to make their own models 
and to discover how easy it is to cut all sorts of regular sections 
through them. For instance: 

1. A cube may be cut into two equal parts by means 



576 CRAFTSMANSHIP IN MATHEMATICS 

of a section of the form of a regular hexagon; the section 
passes through the mid-points of 6 of the 12 edges of the 
cube. 

2. A hexagonal section of an octahedron may be made 
similarly. 

3. A hexagonal section of a dodecahedron, dividing this 
solid also into 2 similar halves, may be made by joining any 
2 opposite angles of any pentagonal face (i.e. by drawing 
any one of the 5 diagonals of any face), and passing a plane 
through this join and the centre of the solid. The 6 edges 
of the hexagon are formed by 6 similar joins. 

If such sections as the last are taken in all possible ways 
through a polyhedron, the intersection of the new planes 
will give clues to all sorts of interesting variations. 

If time can be spared (e.g. after examinations, at the 
end of a Term), good preliminary work can be done in a 
Third or a Fourth Form in preparation for the more formal 
work in the Sixth. We may quote a few sections from Lower 
and Middle Form Geometry: 

Transformations of Polyhedra 

481. Any regular polyhedron may be cut from any other 
polyhedron, and the conversion is always a very simple 
matter. Suppose you cut a vertex (corner) from a cube, 
cutting the three edges back to the same extent; the section 
is an equil. A . Suppose you cut a vertex from an icosahedron, 
cutting all five edges back to the same extent; the section 
is a regular pentagon. Suppose you cut an edge from a cube, 
cutting back the faces symmetrically; the section is a rect- 
angle. This is the sort of thing to be done in the following 
experiments. In one case, all the vertices will be cut off; 
in another, all the edges; always symmetrically and equally. 

The best material to use is best yellow bar soap, from 
which you can easily cut cubes. It is better and cleaner 
than clay or plasticine. (If you are clever with carpenters' 
tools, a fine-grained wood is better still.) For cutting soap 




THE POLYHEDRA 577 

a very thin-bladed knife is desirable. Do not attempt to 
cut off " chunks "; cut off shavings and do the cutting 
gradually. 

482. To cut a tetrahedron from a cube. A cube has 
eight vertices, a tetrahedron has four. The conversion is 
made by cutting away four of the eight vertices; the four 
new planes will be the four faces of the tetrahedron. 

Draw the diag. AC of the top of the cube, and the opposite 
diag. FH in the base of the cube. These are two edges of 
the tetrahedron. Mark them by pushing into the soap three 
or four small pins in each line. Cut off the vertex D, and 
keep paring away until you reach the 
plane ACH, one of the faces of the tetra- 
hedron. Scratch the letters A, C, and 
H on this new plane, or the model may 
become confusing. Now cut off the ver- 
tex G until you reach the plane CFH, 
a second face of the tetrahedron; now 
the vertex B, until you reach the plane Fig. 303 

AFC, a third face; lastly, cut off the 
vertex, E behind, until you reach the plane AFH, the fourth 
face. Now the cutting is completed. Note that the four 
vertices cut off are the four not concerned with the two 
diagonals first drawn. The two vertices at the end of each 
of these diagonals remain, and are the four vertices of the 
new tetrahedron. 

The figure is a little difficult to follow, but the soap 
model itself will keep telling its own story, especially if the 
vertices are all lettered as in the figure. 

483. To cut an octahedron from a cube. This is done 
by cutting off the eight vertices of the cube, the eight new 
planes forming the eight faces of the octahedron. 

Begin by marking in the six vertices of the octahedron. 
They are at " centres " (intersections of the diagonals) of 
the six faces of the cube. Show them by small pins thrust 
into the soap, K, L, M, N, P (and Q in face ABFE, not 
shown). Begin by cutting off the corner C symmetrically, 

(B291) 38 



578 



CRAFTSMANSHIP IN MATHEMATICS 



preserving the equil. A all the time, and pare down until 
you reach the plane made by the three pin-heads K, L, 
and M. KLM is one of the eight faces of the octahedron. 
Cut off the other seven corners of the cube similarly, and 
so obtain the other seven faces. Fig. 304 (ii) shows what 
the model would look like if the eight corners were cut 
down only to the mid-points of the edges. To make the 
octahedron, the cutting has to be continued to the centres 





Fig. 304 

of the faces, by which time the faces of the cube will have 
disappeared. 

484. To cut a dodecahedron from a cube. To 
construct a pentagon, Euclid used golden section. The faces 
of a dodecahedron consist of pentagons, and we shall use 
golden section for the necessary construction in this problem. 

Golden section cuts a 2" line into two parts, <\/5 1 
and 3 \/5 inches long, i.e. 1-24 and -76 inches, very 
nearly. A line 1" long would be cut into parts of -62 and 
38 inches, and a 3|" line into 2-17 and 1-33 inches. We 
will cut the dodecahedron from a 3|" cube. It is a con- 
venient size for handling. If you cannot obtain a piece of 
soap or other material big enough, you must cut a smaller 
cube, but in the same proportion. 

The twelve faces of the dodecahedron are the twelve 
new planes formed by cutting off the twelve edges of the 
cube. Hold a dodecahedron by two opposite edges between 
finger and thumb. These two edges may be regarded as 
medially placed in the top and bottom faces of the cube. 



THE POLYHEDRA 



579 



They are the middle parts of the medians of the faces of 
the cube. The four other edges that occupy medial positions 
in the other four faces of the cube can be easily traced as 
the model is thus held in the fingers. 

The ratio of the length of the edge of the dodecahedron 
to the length of the edge of the containing cube is the ratio 
of the shorter section to the whole line in golden section. 

In a 31" line, the shorter section is 1-33 in. Call this 




Fig. 305 



\\ in. Place a line of this length centrally, as shown, in each 
of the six faces of the cube. (The three not shown in the 
figure they are at the back and underneath correspond 
in position to their opposite neighbours.) These six lines 
we call " medial " lines. 

Fig. 305 (i) shows the construction for each square face. 
Since PQ = 31' and RS = 1|", PR - SQ = l^V". 

The six medial lines of 1J" are six of the thirty edges of 
the dodecahedron. The other twenty-four edges will appear 
when the twelve new planes have been cut. 

Mark the six medial lines by rows of three or four pins 
in each, one pin being placed at each end (G and H for 



580 CRAFTSMANSHIP IN MATHEMATICS 

instance). These can be withdrawn as soon as the cuts nearly 
reach them, in order that the cuts may leave sharp edges. 

Begin by cutting off the edge AB, cutting down to the 
median CD containing MN in the top face, and to the point G 
of the medial line GH in the side face. The new plane is shown, 
shaded. Obtain the other eleven planes in similar fashion, 
every plane extending from one medial to the next, line to 
point, as CD to G, the parallels CD and EF guiding the 
knife. Fig. 305 (iii) shows, separately, another edge cut 
off, and iv shows a third. As one new plane cuts through 
another, a part of the latter will disappear. Confusion will 
arise unless the points in the model are lettered to corre- 
spond with the letters in the figure. 

It is best to cut off four parallel edges of the cube first 
(these four sections will be rectangles), then a second set 
of four parallel edges, then the third. The gradual formation 
of the dodecahedron will then be more clearly understood. 
The pentagons will appear as the last four planes are being 
cut. 

485. To cut an icosahedron from a cube. We might 
proceed as in the last case, first putting into medial position, 
in the six faces of the cube, the longer portion of a line 
divided in golden section, i.e. 2-17 in. in a cube of 3| in. 
edge. If an icosahedron be held by two opposite edges 
between finger and thumb, it is easy to see from the symmetry 
of the solid that these two and four other edges occupy 
medial positions in the six faces of the cube, exactly as in 
the last case. To obtain the twenty faces (equil. As) of the 
icosahedron, we have to cut off not only the twelve edges 
but also the eight vertices of the cube, the new planes by 
their intersections giving the faces. And all the cuts are 
determined by the medial lines in the six faces of the cube. 
But unless the edges, old and new, are carefully marked, 
the cutting is a complicated business. A simpler method 
is first to cut a dodecahedron from the cube ( 484). To 
cut an icosahedron from a dodecahedron is then easy. 

It is done by cutting off the twenty vertices of the 



THE POLYHEDRA 581 

dodecahedron, continuing the cutting until the twenty new 
planes meet to form twenty equil. As. Since each vertex 
of the dodecahedron is made up of three equal plane /.s, 
an equil. A is produced by cutting off a vertex symmetrically. 
Each cut must be continued until the new plane passes 
through the centres * of the pentagons. Hence, with 
twelve pins, mark the centres of the twelve pentagons; they 
show the positions of the twelve vertices of the icosahedron 
to be formed. If you begin by first carrying all the cuts 
to the mid-points of the edges, a very pretty solid is formed, 
consisting of equil. As and small pentagons. But the cuts 
must be carried deeper, until they pass through the centres 
of the original pentagons. 

486. To cut a dodecahedron from an icosahedron. 
This case is almost identical with the last. Cut off a vertex 
of the icosahedron, symmetrically, and a regular pentagon 
results. Cut off all twelve vertices, and twelve pentagons 
result. Continue the cutting until each pentagon passes 
through the centres* of the AS. Hence, with twenty 
pins, begin by marking the centres of the twenty As; they 
show the positions of the twenty vertices of the dodecahedron. 
If you begin by first carrying all the cuts to the mid-point 
of the edges, a pretty solid is formed, consisting of pentagons 
and small equil. As. But the cuts must 

be carried deeper, until they pass through 
the centres of the original equil. As. 

487. To cut a cube from a dode- 
cahedron. Let AB be an edge of a do- 
decahedron, AC, AD two edges, radiating 
from A; BE, BF two radiating from B. If, 

in the solid, CD, DE, EF, FC are joined, FI. 300 

a square is formed, and this is one of the 
faces of the contained cube. It is easy to mark off the other 
five faces, in a similar way, and then to cut out the cube. 
Reductions of this kind lend themselves to useful geo- 

* I.e. the intersection of the J_ bisectors of any two sides. In the case of the 
A , it coincides with the centroid. 




582 



CRAFTSMANSHIP IN MATHEMATICS 



metrical investigations by Sixth Form boys. Here, for 
instance, is an isometrical projection of a cube, showing 
all the construction lines for reducing the cube to an icosa- 
hedron. If the construction lines are based on the 3 initial 
lines ST, VW, XY, the angle at H (138-2) should be 
checked by measurement; and vice versa. The advantage of 
the isometric projection is that all lengths parallel to the 
edges of the cube are true lengths. It is an excellent 




exercise for boys to make cardboard models of the successive 
transition stages of reduction, e.g. the edge AF is first 
removed down to the plane RabH; then the corresponding 
edges through B, C, and D. The face of the model with the 
4: edges removed is then Hbcjde. The other 2 groups of 4 
edges each are treated similarly. Or, if it be decided to remove 
the 8 vertices first, symmetry will decide at once that, e.g., 
the plane cutting off the vertex A must pass through X, T, 
md V, and that Af = Ag Ah. The real trouble comes 
when some of the edges or vertices have been cut away; 
the whole thing then seems to be " lost ". The important 



THE POLYHEDRA 



683 



thing is to keep the 3 initial lines ST, VW, XY in view all 
the time. When a model in soap is made, these initial lines 
may be marked at the outset by thrusting small pins into the 
soap, so that the heads act as guides all through the cutting 
process. 

The following data are useful when effecting polyhedral 
transformations. 



Points and Lines in these containing Polyhedra 


become in the contained Polyhedra 
the following Points and Lines: 


( The 4 Cd's of the 4 F's 
Tetrahedron [ The 4 F's 
1 The mid-pts. of the 6 E's 

( The 4 alt. V's 
Cube \ The 6 D's, i in each F 
I The 6 C's of the 6 F's 

Octahedron. ( The Cd's of the 8 F's 
I The Cd's of the 4 alt. F's 

Icosahedron. The Cd's of the 20 F's 
Dodecahedron. The C. of G. of the 12 F's 


the 4 alt. V's of the cube. 
4 of the 8 F's of the octahedron. 
the 6 V's of the octahedron. 

the 4 V's of the tetrahedron. 
the 6 E's of the tetrahedron. 
the 6 V's of the octahedron. 

the 8 V's /of the cube. 
the 4 V's of the tetrahedron. 

the 20 V's of the dodecahedron. 
the 12 V's of the icosahedron. 



E = edge; F face; V = vertex; C = centre; Cd = centroid; D = diagonal 

A boy who can use the soldering iron may make some 
skeleton models in stout brass wire, fairly deep notches 
being cut with a file symmetrically at each vertex and else- 
where as required. Contained models may be exhibited 
in position by means of stout threads run continuously from 
vertex to vertex, from mid-point to mid-point, and so on. 
When the contained " solid " is completed, the thread is 
brought back to the first vertex and tied. The different 
contained models may be constructed in threads of different 
colours, and each is then easily distinguished from its neigh- 
bours. We showed above how a cube might be cut from 
a dodecahedron: evidently 5 such cubes may be so cut, 
according to the edge we select to begin with. If all 5 cubes 
be threaded in a wire model of the dodecahedron, the re- 
sultant triacontahedron, with its 30 rhombuses, is effectively 



584 CRAFTSMANSHIP IN MATHEMATICS 

shown. With care a 3rd polyhedron may be shown within a 
2nd, and a 4th inside the 3rd, but girls are generally more 
expert than boys in handling and tying the threads. 

The " centres " and centroids of the faces of the initial 
model may be shown by soldering into position diagonals 
or medians of very thin wire. 

A useful present to a keen mathematical boy is Bruckner's 
Vielecke und Vielflache, Theorie und Geschichte (Teubner). 
The many plates contain scores of photographs of beautiful 
models based on the polyhedra. The models are easily 
made in cardboard or stiff paper (I have known excellent 
specimens made by boys during the holidays), and the 
accompanying explanations and theoretical matter are well 
within the range of a Sixth Form boy. Here are reproduc- 
tions of four of the photographs. 



CHAPTER XL 

Mathematics in Biology 

General Ignorance of the Subject 

There is a remarkable ignorance on the part of the average 
person in regard to the numerous matters of mathematical 
interest in botany and zoology. And not all mathematicians 
have interested themselves in this department of their subject, 
even though they may be perfectly familiar with the other 
branches of what they sometimes call " applied " mathe- 
matics. 

The following topics are suggested for inclusion in the 
general mathematical course for Sixth Form non-specialists. 
For such boys only an elementary treatment will be possible, 
though the specialists, if they could spare the time, might 
carry the work much further. Each topic, animal locomotion, 






Fig. 308. Polyhedral Models 



[Facing p. 084 



MATHEMATICS IN BIOLOGY 585 

for instance, is really a very big subject, much too big for 
an exhaustive treatment. But for a boy to leave school entirely 
ignorant of the mathematical significance of the facts enu- 
merated in the following paragraphs is a sad reflection on 
the narrowness of the course of mathematical work that 
schools commonly provide. 

Biological Topics for Consideration 

The principles of similitude in biological forms and 
structures. Why there is necessarily a limit to the size of 
all plants and animals, as well as to all artificial structures. 
Stable equilibrium. How Eiffel secured an even distribution 
of strength in his tower by adopting the form of the logarith- 
mic curve. 

Mechanical efficiency. The stream lines of a fish and 
the lesson to the naval architect; the stream lines of birds 
and the lesson to the student of aeronautics. The human 
skeleton from the engineer's point of view compression 
and tension lines in the construction; ties (ligaments, tendons, 
muscles, &c.) and struts. The structure of a few of the 
principal bones of the body regarded as engineering units, 
and the mechanical distribution of compression and tensile 
stress. Compare the skeletal framework of a quadruped with 
a bridge supported by two piers: the quadruped is really 
an admirably jointed and flexible bridge. Strength and 
flexibility in aquatic animals. The remarkable strength of 
insects in proportion to their size. 

Animal locomotion. Locomotion on land, in air, in 
water. The wing regarded as a helix. The relation between 
the work which a bird does in moving itself forward and 
the linear dimensions of the wings. The minimum necessary 
speed in flight. 

Rate of growth in the organic world. Rate at different 
periods of " life "; its variability and its periodic retarda- 
tion. The weight-length coefficient. (Growth graphs should 
receive special attention.) 



586 CRAFTSMANSHIP IN MATHEMATICS 

Internal forms of organic cells. Fields of force; their 
form; polarity. Effects of surface tension on cell division. 
Liquid films: minimal surfaces and figures of equilibrium. 
Plateau's experiment. Spiders' webs. Forms of globules, 
hanging drops, splashes. Unduloids in the infusoria (e.g. 
vorticella); fluted and pleated cells. 

Cell aggregates. Surfaces in contact; cell partitions; why 
partitions between cells of equal size are plane, and between 
cells of unequal size are curved. Tetrahedral symmetry; 
hexagonal symmetry. The geometry of the bee's cell and 
of bee-cell architecture. Minimal areas in nature's partition- 
ing of space. 

Spicules and spicular skeletons. Concentric striation in 
nature. The fish's age as estimated by the concentric lamel- 
lation of its scales; compare with the concentric rings in the 
trunk of a tree. The skeletons of sponges. The radiolarian 
skeleton. 

Geodesies. The helicoid geodesic on cylindrical structures 
and its purpose; how " stretching tight " and constricting 
are effected by fibres arranged in geodesic fashion. The 
spiral coil in the trachcal tubes of an insect; the tracheides 
of a woody stem. 

The logarithmic spiral. Difference between spirals and 
helices. The curves of the horns of ruminants, of molluscan 
shells, of animals' tails, of the elephant's trunk. The properties 
of the logarithmic spiral in its dynamic aspect. Explain 
clearly why the molluscan shell, like the creature inside it, 
grows in size but does not change in shape: this constant 
similarity of form is the characteristic of the logarithmic 
spiral. The study of shells generally, morphologically, and 
mathematically. The spiral shells of the foraminifera. Torsion 
in the horns of sheep and goats. The deer's antlers. The 
curvature of beaks and claws. 

Phyllotaxis. Spirals. Symmetry. 

Shapes of eggs. An egg, just prior to the formation of 
its shell, is a fluid body, tending to a spherical shape, en- 
closed in a membrane. The problem of the shape of the 



MATHEMATICS IN BIOLOGY 587 

egg: given a practically incompressible fluid, contained in 
a deformable capsule which is either entirely inextensible or 
only very slightly extensible, and which is placed in a long 
elastic tube the walls of which are radially contractile: to 
determine the shape of the egg under pressure. At all points 
the shape is determined by the law of distribution of radial 
pressure within the oviduct; the egg will be compressed in 
the middle, and will tend more or less to the form of a cylinder 
with spherical ends. From the nature and direction of the 
peristaltic wave of the oviduct, the pressure will be greatest 
somewhere behind the middle of the egg; in other words, 
the tube will be converted for the time being into a more 
conical form, and the simple result follows that the anterior 
end of the egg becomes broader and the posterior end the 
narrower. The mathematical statement of the case is simple. 

Comparison of related biological forms. This is a very 
large subject, and applies to the whole region of biological 
morphology. Basically, it consists of the transformation of 
a system of co-ordinates and a comparative study of the 
original and transformed figures (wing, leg, bone, skull, or 
what not) in the co-ordinate system. The new figure in the 
transformed system shows the old figure under strain. The 
new figure is a function of the new co-ordinates in precisely 
the same way as the old figure is of the original co-ordinates. 

The reader should examine figs. 404, 405, and 406 
in Growth and Form (see below), where (i) the outlines of 
a human skull are enclosed in a co-ordinate system of 
squares; (ii) the outlines of a chimpanzee's skull are enclosed 
in another system determined by points exactly corre- 
sponding to the intersecting points in the first system. The 
new co-ordinate system though consisting of curved lines 
is of a strikingly regular type, and obviously bears a very 
simple mathematical relation to the first system. It is for 
the biologist to trace the transformed co-ordinates, for the 
mathematician to step in and show the relations between 
the new and the old co-ordinate systems, and then for the 
biologist to come in again and explain the relations if he 



588 CRAFTSMANSHIP IN MATHEMATICS 

can. That the relations are simple, and that they are con- 
tinuous, are obvious. The logarithmic curve seems to make 
its appearance even once more. (See D'Arcy W. Thompson 
Chapter XVII.) 

The best modern work on the whole subject is Professor 
D'Arcy Thompson's Growth and Form. Professor J. Bell 
Pettigrew's Design in Nature should also be consulted; it 
is a remarkable three-volume work with a multitude of 
useful facts and many hundreds of illustrations, but some 
of its opinions do not meet with general acceptance. 



CHAPTER XLI 

Proportion and Symmetry in Art 

Proportion, Harmony, and Symmetry 

Art is another subject which mathematics gathers into 
its ambit, though the mere suggestion is enough to stir the 
average artist to anger. But what about perspective? The 
subject is generally taught by art teachers as if it consisted 
of a number of incomprehensible stereotyped rules. And 
what about the geometry underlying design? 

But, after all, these things are comparative trifles. Under- 
lying a great deal of what counts for art is a mathematical 
foundation quite unsuspectejjl even by many mathematicians. 

The subject is much too far-reaching for more than a 
few references to it to be made here. 

Those qualities in the general disposition of the parts 
of a building that are calculated to give pleasure to the 
observer are proportion, harmony, and symmetry. In the 
dimensions of a building, proportion itself depends essentially 
upon the employment of very simple mathematical ratios. 
Proportions such as those of an exact cube, or two cubes 



PROPORTION AND SYMMETRY IN ART 589 

placed side by side, or dimensions increasing by one-half 
(e.g. a room 20 ft. high, 30 ft. wide, 45 ft. long), please the 
eye far more than do dimensions taken at random. The 
great Gothic architects appear to have been guided in their 
designs by proportions based on the equilateral triangle. 

By harmony is meant the general balancing of the several 
parts of the design. It is proportion applied to the mutual 
relations of the details. By symmetry is meant general 
uniformity in plan. 

Accurate measurements have been made of the Parthenon 
and of several of the great cathedrals, and the unvarying 
simplicity of the mathematical ratios determining the various 
proportions is a very impressive fact. The same thing applies 
to natural objects. In particular it applies to the human 
figure. An artist does not, of course, measure up a model 
before making a selection, but his eye tells him at once if 
the proportions are satisfactory. If the human body approaches 
anything like perfection, from the crown of the head to the 
thigh joint is one-half the whole height; from the thigh- 
joint to the knee-joint, from the knee-joint to the heel, and 
from the elbow-joint to the end of the longest finger, are 
each one-fourth of the whole height; from the elbow-joint 
to the shoulder is one-fifth; from the crown of the head 
to the point of the chin is one-eighth. The proportions of a 
perfect face are even more remarkable; the ratios of the 
distances between the various facial organs, and of the 
lengths and widths of the organs, are singularly simple through- 
out. 

In great architecture, even more remarkable than the 
linear measurements is the simplicity of the forms of the 
various rectilinear and curvilinear spaces. If we analyse 
a drawing of, say, the east front of Lincoln Cathedral, we 
can discover a series of striking relations amongst the parts. 
First enclose it in a rectangle, and then draw the bisecting 
vertical line; from the upper end of that vertical, draw 
7 pairs of oblique lines, right and left, to the base and sides 
of the rectangle, as follows: 



590 CRAFTSMANSHIP IN MATHEMATICS 

1. Lines (to the angles) which determine both the width 
of the design, the tops of the aisle windows, and the bases 
of the pediments on the inner buttresses. 

2. Lines which determine the outer buttress. 

3. Lines which determine the width of the great centre 
window. 

4. Lines which determine the form of the pediment of 
the centre. 

5. Lines which determine the form of the pediments of 
the smaller gables. 

6. Lines which determine the height of the outer but- 
tress. 

7. Lines which determine the height of the inner but- 
tresses 

It will be found that (a) these lines determine the heights 
and widths of nearly all the main features of the design, 
and (j8) the angles which the obliques make with the hori- 
zontal are all simple fractions of a right angle. Were the 
architect to depart from these simple ratios very appreciably, 
the eye would be offended; the trained eye would resent 
even a very small departure. 

Ratio Simplicity 

The key to the harmony of beauty in its more general 
sense seems to be the simplicity of the angle relations which 
determine or which underline the form of the thing con- 
sidered beautiful. 

It is sometimes said that there are three primary "orders" 
of symmetry, viz. those based on the numbers 2, 3, and 5, 
respectively. That of the first order is represented by the 
half square cut off by a diagonal, that is, the right-angled 
isosceles triangle, with angles 45, 45, 90 (angles 1:1: 2); 
that of the second order is represented by the half equilateral 
triangle cut off by a median, that is, a triangle with angles 
30, 60, 90 (angles 1:2:3); that of the third order is 



PROPORTION AND SYMMETRY IN ART 



591 



represented by the half triangle from a pentagram cut off 
by a median from the vertex, that is a triangle of 18, 72, 
90 (angles 1:4:5). Thus we may easily obtain angle 
ratios 1/1, 1/2; 1/2, 1/3, 2/3; 1/4, 1/5, 4/5; and these ratios, 
alone or compounded in a simple way, are architecturally 
fundamental. 

An ellipse, a figure which enters very largely into archi- 
tectural composition, may be constructed with its principal 
diameters of any ratio, but the ellipses which are acceptable 
for purposes of symmetrical beauty are those based upon 
the simplicity of the ratio of the 
angles made by the diagonal with 
the sides of the rectangle which 
encloses the ellipse. This ratio is 
usually one of those mentioned in 
the preceding paragraph. 

But a much more subtle curve 
is the " oval ", better called a 
" composite ellipse ", since its axis 
is sometimes so short that the oval 
ceases to resemble an egg. This 
curve has 3 foci (A, B, C in figure), 
forming an isosceles triangle. If 
from the ends of the base of this 

triangle lines be drawn to that end of the axis, D, at the 
" flatter " end of the curve, they make an angle which (for 
the purposes under consideration) must have a very simple 
relation to the angle at the apex of the isosceles triangle 
(in the figure, 3 : 1). If this simplicity of ratio is departed 
from, the curve is not acceptable as an element of har- 
monious proportions. 

It is a remarkable thing that a form is considered beautiful 
when the space which it encloses can be analysed in such a 
way that the resulting angles bear proportions to each other 
analogous to those which subsist among musical notes. 
The basis of musical harmony is that, when two sounds 
mingle agreeably, the numbers of vibrations of which they 




Fig. 309 



592 



CRAFTSMANSHIP IN MATHEMATICS 



are respectively composed bear a very simple ratio to each 
other. All the harmonies are represented by quite simple 
fractions, J, , , &c. 



Some things to look for in objects considered 
beautiful 

Boys should be encouraged to take an interest in the 
proportions, harmony, and symmetry of beautiful buildings 
and other objects. I have known many pupils who claimed 
to be beauty-blind, really awakened to a new life once they 
knew what to look for when examining a thing considered 
beautiful a building, a piece of statuary, a picture, a vase, 

a piece of Gothic orna- 
ment. 

Search for the beauty 
of form of a Greek or 
Etruscan vase: it does not 
take much finding. Stand 
in front, or behind, or at 
the side, of the Venus of 
Milo in the Louvre; the 
particular view matters 
little. The extraordinary 
beauty of the curves of 
the figure, despite their 
complexity, imposes itself 

Fig. 310 upon the mathematician 

whether he will or not. To 

the artist the figure is beautiful for reasons which, though ade- 
quate, he finds a little difficult to explain, or at least difficult 
to analyse. The mathematician's approach is an entirely 
different approach, but the approach is intensely interesting; 
he desires to discover the secret of the artist's construction, 
and he sets to work to analyse. Once the artist's secret 
stands revealed, his first feeling is one of admiration for 
such subtle craftsmanship. The beauty of the thing gradually 




PROPORTION AND SYMMETRY IN ART 



693 




Fig. 3 1 1 

grows upon him. The feeling is as much intellectual as 

it is emotional, and for that reason it is sometimes of a higher 

order than a feeling for beauty that may be 

emotional only. The boy who is a failure in 

the school studio can nevertheless be taught 

what to try to search for in a thing that the 

world calls beautiful. 

We give figures of (1) an Etruscan vase 
showing its component curves and their tan- 
gential relations; (2) the construction of the 
echinus moulding in Greek architecture (note 
the 3 composite ellipses); (3) the outline of 
the human figure, showing nature's subtle 
construction of the sides of the head, neck, 
trunk, and outer surfaces of the legs. Note 
the tangency throughout. In an analysis of 
a perfect human form, these tangential rela- 
tions seem to persist to the smallest detail. 
(Figs. 310, 311, 312.) 

If one of the ends of life is the pursuit 
of beauty, then mathematics, properly under- 
stood, is one of the avenues we should 
follow. Consider even Einstein's work; what 
is its main value? that> underlying the diverse 

(K291) 




594 CRAFTSMANSHIP IN MATHEMATICS 

phenomena of the natural world, there has been dis- 
covered a harmony more all-embracing than any ever before 
dreamed of. 

I do not know if it may be adequately maintained that 
harmony is the most essential factor in beauty, but assuredly 
it is the desire for harmony that animates the modern searcher 
after the secret of the ordered relations of the universe, 
just as it animated the Greeks in their star-gazing and their 
geometry. 

I find it very hard to distinguish the passion for truth 
from the quest of beauty. Certainly we need never despair 
of the beauty-blind boy if he is taught mathematics as it 
might be taught. 



CHAPTER XLII 

Numbers : Their Unexpected Relations 

The Theory of Numbers 

One noteworthy subject which is lacking in the equip- 
ment of many of our younger mathematical teachers is the 
theory of numbers. Forty or fifty years ago the subject was 
included as a matter of course in Sixth Form mathematics. 
In those happy days the mathematical work was not narrowed 
down to the requirements of the few boys who were going 
to read mathematics at the University. The ground then 
covered was more extensive, and in many ways was more 
interesting. Some schools devoted considerable attention to 
the theory of numbers (as the subject is called, though not 
very happily), such topics being included as the theory of 
perfect, amicable, and polygonal numbers; properties of 
prime numbers; possible and impossible forms of square 
numbers, of cubes, and of higher powers; the quadratic 



NUMBERS: THEIR UNEXPECTED RELATIONS 595 

forms of prime numbers; scales of notation; indeterminate 
equations; diophantine problems. The magic square and 
magic cube were also included. Altogether, the boys were 
given an interest in arithmetic and algebra that remained 
a permanent possession in after days, and was rarely forgotten. 

A final blow was given to this branch of work when the 
fiat went forth that circulating decimals being useless and 
unpractical, their use must be abandoned. The futility of 
circulating decimals in the solution of practical problems 
may be granted. But if we ignore them altogether, we cut 
off from the learner some of the most striking properties of 
numbers; in fact we deny him most of the inner significance 
of numerical relations. 

I plead for a revival of some of this work, and therefore 
indicate a few topics that may be included. 

Within the last year or two, a highly competent young 
mathematical mistress told me that she had made what 
she considered to be a rather striking arithmetical discovery. 
Had I come across it before? 

This is what she showed me, and then she pointed out a 

\ = -i4285t 
| == -285714 
? - 428571 
-57142 



J = -714285 



f = -57145 

few of the well-known properties of this particular grouping: 
the same 6 figures in the same order in all 6 cases; the 1st 
group a factor of all the others; the 2nd group a factor of 
the 4th and 6th; the 5th group the sum of the 2nd and 3rd; 
and so forth. She said quite seriously that although she 
had, as a girl, won an open scholarship, she had never during 
her school days seen the completed decimal for any one 
of these half-dozen simple fractions. 

I pointed out that she had not made up the group to the 
best advantage, and I modified it thus: 



596 



CRAFTSMANSHIP IN MATHEMATICS 





i 

T 


3 


f 


n 

r 


* 


K 

T 


1 


1 


4 


2 


8 


5 


7 


3 


4 


2 


8 


5 


7 


1 


2 
T 


2 


8 


5 


7 


1 


4 


6 

T 


8 


5 


7 


1 


4 


2 


4 

T 


6 


7 


1 


4 


2 


8 


5 

r 


7 


1 


4 


2 


8 


5 



The values of the 6 vulgar fractions may now be read off 
either from left to right or from top to bottom. Moreover, 
the diagonal lines running upwards from left to right consist 
each of the same figure. 

Even then she could scarcely credit my statement that 
the same principle applied to all circulating decimals what- 
soever. I suggested she should evaluate the 40 fractions 
4~O ITD 4~f t J i > which she did, and made the discovery 
that the 40 circulating decimals fell into 8 groups of 5 figures, 
all presenting the self-same symmetry. 



1 

TT 


= -6243$ 


Tl 


0487 


3 
4T 


073lt 


4 
If 


09756 


1 

TT 


= -24396 


20 
IT 


48780 


4 1 ~ 


17073 


23 
41 


5609? 


l fl 
4T 


= -59024 


4 T ~ 


7804& 


13 
4 1 


3170? 


2 5 
41 


60975 


18 
41 


= -43902 


:) 3 
4 \ 


0487 


2 
i 1 


70731 


.'51 
41 


75609 


3 7 
TT 


= -&)243 


4 T ~ 


7804 


:io 
4 T 


73176 


40 
41 


&7560 


fi 
IT 


= -12195 


C, 
41 ~ 


14634 


i i 

41 


2682$ 


if ~ 


36585 


8 
"41 


= -19512 


14 
41 


54146 


1 li 
41 


2926 


"4T ~ 


5365 



TT 


= -21951 


17 
4T 


41465 


28 
41 


68292 


2 4 
TT 


58536 


21 
4 1 


= -51219 


10 
41 


46341 


84 
41 


2926 


27 
41 


65853 


39 
41 


= -95121 


20 
ll 


63414 


38 
41 ~ 


&26S2 


3 5 
TT 


5365 



Make the boys evaluate these or other similar groups, 
and encourage them to search for the curious (though obviously 
necessary) relations between the members of each group 
and between group and group. For instance, the first decimals 
of the above 6 groups are 1:2:3:4:5:6; the sum of the 
1st decimals of the 5th and 6th groups is equal to the first 
decimal of the 7th group; and so on almost indefinitely. An 



JNUMBERS: THEIR UNEXPECTED RELATIONS 597 



examination of the numerators of the vulgar fractions gives 
the clue to an almost endless number of relations amongst the 
40 decimal groups. Clearly all 40 decimals can be evaluated 
in 5 minutes; only one actual division is necessary, viz. that 
for -^Y. This sort of thing, which applies universally, was 
the A B C of Upper Form arithmetic half a century ago. 

Suggested Topics 

1. Primes and composite numbers; measures and multiples; 
tests of divisibility (with algebraic proofs). Familiarity with 
the factors of such common numbers as 1001 (=7 X 11 X 13), 
of 999 (= 27 X 37), in order to write down at once the factors 
of such numbers as 702,702 and 555,888. Eratosthenes 1 sieve. 
Fermat's theorem. The number of factors in a composite 
number; the number of ways in which a composite number 
may be resolved into factors. 

2. Perfect numbers. (A perfect number is one which is 
equal to the sum of all its divisors, unity included). N = 
2 n ~ 1 (2 n 1), the bracketed factor being prime. Examples: 28, 
496,8128. 

3. Amicable numbers. (Amicable numbers are pairs of 
numbers, each member of a pair being equal to the sum of 
all the divisors of the other number.) Examples: 220 and 
284; 18,416 and 17,296. The formulae are rather long though 
easy to manipulate 



n 12 






I 5 12 22 



Fig. 313 



4. Polygonal numbers. Teach the pupils (1) to sum 
triangular, square, pentagonal, and hexagonal numbers, and 



598 CRAFTSMANSHIP IN MATHEMATICS 

(2) to find the ?ith term of each series. All are easy and are 
full of interest. The necessary figures may be drawn readily. 
Pascal's arithmetical triangle. 

5. Scales of notation. Illustrate by some of the mediaeval 
problems on age-telling cards; weighing with a minimum 
number of weights, e.g. binary scale weights 1, 2, 2 2 , 2 3 , 
&c., for one pan; ternary scale weights 1, 3, 3 2 , 3 3 , &c., 
for either or both pans; &c. 

6. Congruences. Use Gauss's notation a = b (mod. m)\ 
e.g. 15-8 (mod. 7), 36 - (mod. 12), 37 = 19 (mod. 6). 
Emphasize the fact that a modulus is a divisor. The numbers 
15 and 8 are congruent, or they agree, for the modulus 7, 
because they agree as regards the divisor 7; they " agree " 
in giving the same remainder, 1. The theory of congruences 
is necessary for a proper understanding of parts of the theory 
of numbers. Very little practice in a few of the more ele- 
mentary theorems of congruences is required in order to 
give necessary facility in subsequent work; e.g. 

(1) 72 = 37 = 30 =- 9 = 2 = -5 = -12 = -33 (mod. 7). 

(2) 100 = 15 (mod. 17). 
i.e. 10 2 = 15 (mod. 17), 

.'. 10 4 = 15 2 (mod. 17), 
.'. 10 4 = 225 (mod. 17), 
/. 10 4 = 4 (mod. 17). 

From the 225 we have subtracted 13 times the modulus and 
have thus brought it down to a number smaller than the 
modulus. The pupils should be familiarized with this prin- 
ciple. 

7. Circulating decimals. How many of our younger 
readers know how easy it is to write down, almost at top 
speed, the complete circulating decimal equivalent to a 
vulgar fraction in its lowest terms, if the denominator is 
prime, no matter how many figures the period may consist 
of? The following section will suffice for a general introduc- 
tion to this interesting subject. 



NUMBERS: THEIR UNEXPECTED RELATIONS 599 



Circulating Decimals and Congruences 

We will set out below the complete evaluation of the 
decimal corresponding to such a fraction, say -^. This gives 
a recurring period of 28 places, and we shall therefore write 
down 1 followed by 28 ciphers and divide by 29 in the usual 
way. We will choose short division in order to show the 
successive quotient figures and their corresponding re- 
mainders clearly. Over the ciphers we will write the successive 
numbers 1 to 28, in order to be able to refer at once to any 
particular quotient figure or to any particular remainder. 
We will call the quotient figures, Q's, and the remainder, R's. 
Examine the 28 R's carefully: they consist of all the numbers 
1 to 28, the last of them being 1; this typifies all evaluations 
of circulating decimals 



29 )1-000000000000 0000000000000000 
Q ' s ~* HSliTTs 27586 2"06 8965517241 37 9 3~1 

R s _+ ,. rrn; . 2 , i, * 5 i, . , 20 ,. 28 inrrirT i. * - .rrrr 

(It may be observed that the unit figure of each R is identical 
with the corresponding Q; but this is not universal; it 
occurs only when the main divisor has 9 for its unit figure.) 
The pupils' interest may readily be excited in this way: 
Twenty-nine is a rather hard number to divide by. Can 
we substitute a smaller and easier number? Yes, at any 
rate after we have found the first few figures (Q's) of the 
answer by dividing by the 29, say as far as Q n , i.e. when 
we have '03448275862. Now begin again at the begin- 
ning and divide this part-answer by 5 (ignore the decimal 
point for the present). 

5)3448275862 
^0689655, &c. 

But these figures 0689655, &c., are the figures beginning at 
Q 12 in the answer, and we may continue to divide by 5 



600 CRAFTSMANSHIP IN MATHEMATICS 

until we reach the end. But note that we no longer bring 
down O's, but the Q's we have previously written down. 

But we need not have divided by 29 so far as Q n . Suppose 
we had gone as far as Q 9 . We write down as before the 
figures thus obtained (omitting the decimal point and the 
which follows it), prefix a 4, and then divide by 7. 

7)^34482768 
""62068, &c. 

But these figures 62068, &c., are the figures beginning at Q 10 
in the answer, and we may continue to divide by 7 until 
we reach the end. In this case we have shortened still more 
the original division by 29. 

But we can shorten the division by 29 still further. Sup- 
pose we had proceeded as far as Q 3 , and had obtained -034. 
As before we write down the figures thus obtained (omitting 
the decimal point and the cipher), prefix a 4, and then 
divide by 9. 

9) 4 34 
_48, c. 

But these figures 48, &c., are the figures beginning at Q 4 
in the answer, and we may continue to divide by 9 until 
we reach the end. Thus we may begin to divide by 9 after 
obtaining only 3 figures by dividing by 29. 

Note carefully how we proceed with the division in the 
last case. We had, to begin with, -034. To proceed with the 
division we prefix a 4 to the 3, put each new Q in its proper 
place, and remember to " bring it down " (not bring down 
a 0) when its turn comes. 

9's into 43 = 4 and 7 over. Hence -0344. 

9's 74 = 8 2 -03448. 

9's 24 = 2 6 -034482. 

9 s 68 = 7 5 -0344827. 

Really, however, it is unnecessary to divide by 29 more 
than 2 places, i.e. when we have obtained Q x and Q 2 (= -03). 
This time we prefix to the Sal, and thus obtain 13 for our 



NUMBERS: THEIR UNEXPECTED RELATIONS 601 

initial bit of new dividend. This time we can use the easy 
divisor 3. 

3's into 13 = 4 and 1 over. Hence -034. 
3>s 14 = 4 2 -0344. 

3>s 24 = 8 -03448. 

3' s 08 = 2 2 -034482. 

3's 22 = 7 1 -0344827. 

Observe carefully in this case that, at each step, the new 
quotient figure and the over figure give, reversed, the number 
to be divided at the next step; e.g. 4 and 1 over in the first 
line give 14, the number to be divided in the second line. 
And so generally. 

To divide by 3 is so easy that we may evaluate the whole 
28 figures of the period in half a minute. 

But we need not divide at all. We may begin at the other 
end, and multiply instead. 

Suppose we know the last 5 figures, . . . 37931 (Q 24 
to Q 28 ). We may multiply the end figure 1 by 11 and obtain 
Q 23 , and then proceed in this way: 

IXll = 11; 1 down (= Q 23 ) and 1 to carry. 

Hence 137931. 

(3 x 11) + 1 = 34; 4 down (= Q 22 ) and 3 to carry, 

Hence 4137931. 

(9 X 11) + 3 = 102; 2 down (= Q 21 ) and 10 to carry, 

Hence ..24137931. 

But other multipliers might be used. Suppose, for instance, 
we know only the very last figure, 1 (= Q 28 ). This is quite 
enough, if we use the multiplier 3, and we may finish the 
whole thing in a few seconds. 

1x3 =3 31. 

3x3 =9 931. 

9x3 =27 7931. 

(7 x 3) + 2 = 23 37931. 

(3 x 3) + 2= 11 ...137931. 

A similar scheme applies universally. Whence the secret? 



602 CRAFTSMANSHIP IN MATHEMATICS 

The secret lies in the R's and in the Q's, and in the use of 
congruences. 

Give the pupils a ten minutes' lesson on congruences. 
The Sixth Form specialists are almost certain to hit upon 
the solution. The following hints ought in any case to 
suffice. 

Consider another example: 



17) 10000000000000000 
Q's -> 0588235 2 oTTT? 647 
R's -> " " 1 i. . . . >. n . i 

The divisor 17 may be regarded as the modulus of a 
congruence. 

(i) 10 1 = 10, 

.-. (10 1 ) 2 = 10 a , i.e. 10 2 = 100 - 15 = R 2 . 

(ii) 10 l = 10, and 10 2 = 15; 

/. 10 1 .10 2 = 10.15, i.e. 10 3 = 150 = 14 = R 3 . 

(iii) 10 2 = 15, and 10 3 = 14; 

.'. 10 2 .10 3 = 15.14, i.e. 10 5 = 210 = 6 = R 5 . 

(iv) 10 3 = 14, and 10 5 = 6; 

/. 10 3 .10 5 EE 14.6, i.e. 10 8 = 84 = 16 = R 8 . 

All these results agree with the actual division. Clearly, 
then, when the first R has been found by actual division, 
any other remainder whatsoever may be found by applying 
the principles of congruences to the powers of 10. 

It thus follows that once we have detected a multiple 
relation between a pair of R's, the multiple may be used as 
a general divisor for obtaining Q's, and actual dividing in 
the original division need be carried only a very little distance 

Consider another example: 



59)1-000000000000, &c. 
Q's-> 16949152542,~&T. 

n >g ^ 10 41 it 29 64 <1 1C IS It iTfl C- 

An examination of the R's shows that R 5 = 6R 6 . This 
gives us the clue to the relation between every pair of succes- 



NUMBERS: THEIR UNEXPECTED RELATIONS 603 

sive R's, and therefore to every pair of successive Q's. For 

instance, 

R 5 = 6R 6 (54 - 6 . 9) 

R 6 -f 3M = 6R 7 (9 + 177 = 6 . 31) 
R 7 + M = 6R 8 (31 + 59 = 6 . 15) 
R 8 + 3M - 6R 9 (15 -f 177 - 6 . 32), 

where the multiple relation suggests the divisor 6. 

Consider the first of these, R 5 = 6R 6 . If we have pro- 
ceeded with the division as far as Q 6 , we divide Q 6 by 6, 
and so obtain Q 7 (= 1) and 3 over; 6's into 31 = 5 and 
1 over; &c. 

Or, consider the last of the 4 relations: R 8 + 3M = 6R 9 . 
If we have proceeded with the original division as far as 
Q 9 , we prefix a 3 (representing 3M) to Q 9 (= 2), making 
32, divide by 6, and so obtain Q 10 . 

Or, consider the third of the 4 relations: R 7 + M = 6R 8 . 
If we have proceeded with the original division as far as Q 8 , 
we prefix a 1 (representing 1M) to Q 8 (=5), making 15, 
divide by 6, and so obtain Q 9 . 

We may begin to divide by 6 at any point after Q . It 
is merely a matter of prefixing a figure indicated by x in 



In our first example, - 2 ^, we first divided by 5, because 
we noted the relation R x == 5R n ; then we divided by 7, 
because we noted the relation R 3 7R n ; then we divided 
by 9, because we noted the relation R 9 = 9Rn- An easy 
divisor can always be obtained by examining the R's in this 
way. We finally divided by 3 because we noted that R 10 = 3R U . 

But with a little practice we can dispense with the R's 
altogether, and detect a multiple relation amongst the early 
Q's, these being obtained, of course, by actual division. 
For instance 

^ = -6434782608695652173913. 

After obtaining about 5 Q's, we might notice that by pre- 
fixing 2 to Q 2 , making 24, we might divide by 7 and obtain 
Q 3 , and so on continuously. 



604 CRAFTSMANSHIP IN MATHEMATICS 

Here is an example for the boys to complete. The recurring 
period consists of 646 places. A good Sixth Form boy 
ought to write down 1 figure per second, and so do the whole 
thing in 10 or 11 minutes. 

F J T = -601545595 . . . 057187011 

A suitable divisor or multiplier is 11. The division by 11 may 
be begun after the first 3 places are obtained (-001) by 
prefixing 6 to Q l and so obtaining 6001. (A smaller divisor 
may soon be found.) 

11 's into 60 = 5 (= Q 4 ) and 5 over; 
ll's 50 = 4(^Q 5 ) 6 ; 
ll's 61^5(=Q 6 ) 6 ; 
ll's 65-5(=Q 7 ) ,,10 ; &c. 

In the multiplication, 

7 (= Q 6 4 6 ) X 11 - 77; 7 (= Q 643 ) and carry 7; 
{ 1 (= Q 6 4 5 ) X 11 } + 7 = 18; 8 (= Q 642 ) and carry 1, &c. 

The result is easily checked by selecting other divisors. 

If the numerator of the vulgar fraction is other than 
unity, the equivalent decimals will consist of the same figures 
as when the numerator is unity, and they will be in the same 
order, but the period will begin in a different place, easily 
discovered. But the subject, which is a source of delight 
to most boys, cannot be carried further here. 



Magic Squares and Magic Cubes 

Can time spent on this subject be justified? If as a 
mathematical topic for purposes of formal teaching, no. If 
as a subject for creating a lasting mathematical interest in 
the less mathematically inclined boys, yes. 

Magic squares have interested the greatest mathematicians. 
The wonderful harmony and symmetry of the numbers so 
grouped have always tended to attract their attention. Boys 
are always impressed by the mysterious regularity that 



NUMBERS: THEIR UNEXPECTED RELATIONS 605 



emerges in so many ways when they study magic squares. 
Three or four lessons on the subject are well worth giving, 
though in so short a time the secrets of the construction of 
some of the remarkable squares that have been constructed 
by mathematicians could not be given. 

The simplest and best-known construction (for a square 
with an odd number of sides) is the following: the method 











5 










4 




10 


3 




9 




15 


2 




8 




14 




20 


1 




7 




13 




19 




25 




G 




12 




18 




24 






II 




17 




23 






16 




22 






21 





3 


16 


9 


22 


15 


20 


8 


21 


14 


2 


7 


25 


13 


1 


19 


24 


12 


5 


18 


6 


II 


4 


17 


10 


23 



e> 



c 



Fig. 3H 

underlying the symmetrical transfer of the numbers in the 
temporary outer cells is obvious. Horizontally, vertically, 
and diagonally, the sum of the numbers is 65. 

Paste such a magic square round a roller, the circumference 
of which is equal to AB or BC. Two squares should be pre- 
pared, one to be rolled round from side to side, so that AB 
coincides with DC, and one from top to bottom so that 
AD coincides with BC. The consecutive numbers in the 
various diagonals of thfe rolled up squares give the learner 
the real secret of the construction of the simpler squares. 

The famous Benjamin Franklin was the inventor of magic 
squares with properties that always fill the learner with 
astonishment. The construction of his 8 X 8 and 16 X 16 
squares is quite simple, and this very simplicity goes far 
to create the astonishment. They are to be found in all the 
textbooks on the subject, and every boy should know them. 



606 



CRAFTSMANSHIP IN MATHEMATICS 



The problem of finding the number of different ways 
in which the numbers, say, 1 to 25, may be arranged in a 
square is worth looking into, though no general solution 
has yet been discovered. Boys find out at once, of course, 
that variations are easy to make and are numerous. 

Not all the textbooks point out the device for making 
magic squares so that the products of the columns and rows 






Fig. 315 

are constant. It is simply a question of using the numbers 
in the ordinary squares as indices of some selected number 
for the new square. We append the usual 3x3 square 
(common sum 15) and one of its cousins (common product 
= 2 15 = 32768). 

" Magic cubes " may be touched upon, say 3 X 3 X 3 
(1 to 27). The sum of the numbers in each row is 42, not 




only in each face shown, but through the faces, front to back; 
also the diagonals of the cube as well as the diagonals of 
some of the squares. 

Magic circles, pentagrams, &c., are hardly worth spending 
time over. 

The best books on the subject are (1) Magic Squares and 



NUMBERS: THEIR UNEXPECTED RELATIONS 607 

Magic Cubes, John Willis: (2) Magic Squares and Cubes, 
W. S. Andrews; (3) Les Espaces Arithmdtiques Hypermagiques, 
Gabriel Arnoux; (4) Les Carres Magiques, M. Frolow; 
(5) Le Probleme d'Euler et les Carres Magiques, Atlas, M. 
Frolow. 

Magnitudes. Great and Small 

When attempting to help a boy to form a clearer conception 
of the significance of very large numbers, say those con- 
cerning stellar distances or atomic magnitudes, it is essential 
for the teacher to eliminate from the problem every kind 
of avoidable complexity. To form a conception of a great 
number is quite difficult enough in itself, and to a boy the 
difficulty may prove insuperable. On one occasion I heard 
a teacher attacking our old friend the " light-year ", in 
favour of its new rival the " parsec ", simply on the ground 
that the latter made astronomers' computations easier. Now 
the light-year is a perfectly well understood thing. In 
mechanics we often define distance as the product of velocity 
and time (s = vt), as every child knows; and we apply 
this self-same principle to the distance known as a light- 
year, the new unit being determined by the product of the 
velocity of light (miles per second) and the number of 
seconds in a year. But the parsec is the distance corre- 
sponding to the parallax of 1", and a simple calculation 
shows that it is 3*26 times as long as the light-year; and 
this trigonometrical method of determining star distances 
compels the learner to think in terms of the semi-major 
axis of the earth's orbit. The complexity is entirely un- 
necessary in school work; it tends to obscure the main 
thing the boy is supposed to be thinking about. 

If the learner already knows that the velocity of light 
is 186,000 miles a second, simple arithmetic tells him that 
the length of the light-year is, approximately, 

(186,000 X 60 x 60 X 24 x 365) miles, i.e. 6 x 10 12 miles, 
or 6 billion miles. Thus, when the learner is told that a Cen- 



608 CRAFTSMANSHIP IN MATHEMATICS 

tauri is 4 light-years distant, he knows that this means 24 
billion miles; and that the 1,000,000 light-years representing 
the probable distance of the remoter nebulae is a distance 
of 6 X 10 18 (six trillion) miles. Or, he may be told that 
the mass of the H atom is 1-66 X 10~ 24 grams, when he 
sees at once that 10 24 H atoms together must weigh 1 grams. 

But are these vast numbers anything more than mere 
words to the boy? What does a quadrillion signify to him? 
or even a trillion or a billion? or even a million? Is it of 
any use to try to make the boy realize the significance of 
such numbers? or just to leave them as mere words? or 
not to mention them at all and merely to give some such 
illustration as Kelvin's earth-sized sphere full of cricket- 
balls? 

I have tried the experiment of giving to boys such illus- 
trations as these: (1) the number of molecules in 1 c. c. of 
gas is about 200 trillions (2 X 10 20 ), a number equal to the 
number of grains of fine sand, 70,000 to the cubic inch, 
in a layer 1 foot deep, covering the whole surface of England 
and Wales; (2) the number of molecules in a single drop of 
water is about 1700 trillions (1-7 x 10 21 ), a number just about 
equal to the number of drops of water in a layer 7-| inches 
deep completely covering a sphere the size of the earth. 
But I have always found that such illustrations merely 
give rise to vague wonderment. The pupil himself makes 
no personal effort to realize the magnitude of the numbers; 
and this is fatal. 

Such an effort is indispensable. The best plan, perhaps, 
is to make the pupil first consider carefully the magnitude of 
a million, then of a billion, a trillion, a quadrillion, successively 
(10 6 , 10 12 , 10 18 , 10 24 ). For instance, an ordinary watch ticks 
5 times a second or 1000 times in about 3 minutes, or a 
million times in about 2 days and 2 nights. Let this fact be 
assimilated as a basic fact, first. Now let the boy think about a 
billion. Evidently, a watch would take (for the present 
purpose, all the underlying assumptions may be accepted) 
about 6000 years to tick a billion times (2 days X 10 6 ), so 



NUMBERS: THEIR UNEXPECTED RELATIONS 609 

that if a watch had started to tick at the time King Solomon 
was building the Jewish temple, it would not yet have ticked 
half a billion times. Now proceed to a trillion, and then 
to a quadrillion. Evidently the watch would take 6000 
million years to tick a trillion times, and 6000 billion years 
to tick a quadrillion times. An approach of this kind to 
the subject does not take long, and a boy fond of arithmetic 
may be encouraged to invent illustrations of his own. It 
is worth while. It is worth one's own while, if the attempt 
has never been made before. It is, indeed, hard to realize 
the significance of the statement that light-waves tap the 
retina of the eye billions of times a second. Yet how are 
we to escape accepting this frequency if we accept the measured 
velocity of light and the measured length of light-waves? 
Impress upon the boys the fact that the inference is inescap- 
able. 

The description of the manufacture of such a thing as a 
diffraction grating with lines ruled 20,000 to the inch, or of 
Dr. J. W. Beams* mechanical production of light flashes of 
only 10~ 7 second duration, serves to impress pupils with the 
sense of reality of small things. 

Boys must understand that both stellar magnitudes and 
atomic magnitudes are, for the most part, calculated values 
and not directly measured values, and that the calculations 
are, in the main, based on inferential evidence, the inferences 
being drawn partly from known facts, partly from hypotheses. 
But converging evidence of different kinds justifies a feeling 
of confidence in the probability of the truth of the estimate. 
So much so is this the case, that the natural repugnance of 
the mind to accept statements which seem to be so contra- 
dictory of everyday experience, and therefore to " common 
sense ", is overcome. Still, the nature of the evidence avail- 
able must be borne in mind. So must the amazing nature of 
the results. 



(E 291 ) 40 



610 CRAFTSMANSHIP IN MATHEMATICS 

The same subject. Sir James Jeans 9 Methods 

Sir James Jeans, in his two recent books, The Universe 
Around Us and The Mysterious Universe, has adopted 
various devices for helping the mathematically uninitiated 
to realize the significance of large numbers. The first book 
is " written in simple language " and is intended to be 
" intelligible to readers with no special scientific attain- 
ments ". The second book " may be read as a sequel ". 

In order to find out how the books might appeal to some 
of their readers, I induced seven non-mathematical friends 
(two specialists in Classics, two in History, one in Modern 
Languages, two in Science) to read the books through and 
then submit to be questioned on the meaning of the following 
(and a few other) extracts: 

1. " Less than a thousand thousand millionth part." 

2. " 15 million million years." 

3. " 2000 million light-years. " 

4. " The nearest star, Proxima Centauri, is 25,000,000 
million miles away." 

5. " An average star contains about 10 56 molecules." 

The chemist was on the spot at once. The biologist had 
to do a good deal of thinking, but he got there at last. But 
the other five? They failed utterly to understand what the 
numbers signified, though all when at school had taken 
mathematics in the School Certificate (or its equivalent), 
and one in the Higher Certificate. " What is the difference 
between ' 15 million million years ' and ' 15 million years '?" 
" Oh, the former means just a few more millions than the 
latter, I suppose." To the Higher Certificate man I said, 
" Compare the increase of 10 4 to 10 7 with the increase 10 53 
to 10 56 ." He replied: " It is exactly the same thing, for in 
each case you have increased the index by 3. If you take 
10 4 from 10 7 you get 9,990,000, so that if you take 10 53 from 
10 56 the difference must be just the same "! And so generally. 
The five examinees had but the vaguest notions of what the 



NUMBERS: THEIR UNEXPECTED RELATIONS 611 

numbers meant. Finally they all admitted that, elementary 
as the mathematics of the books appeared to be, they simply 
did not understand it. The books had left in their mind 
feelings of intense wonderment, but the real facts they had 
not grasped at all. It would be interesting if other teachers 
would test some of their own friends similarly. 

I am not sure that we gain anything by writing "15 
million million " instead of " 15,000,000,000,000 ", or " 15 
billion ", or " 15. 10 12 ". Are the words " million million " 
significant? 

Some of Sir James Jeans' illustrations are well worth 
examining. Here is one: 

1. Age of telescopic astronomy, 300 years. 

2. Age of astronomical science, 3000 

3. Age of man on earth, 300,000 * 

4. Age of life on earth, 200,000,000 

A teacher can make much of such a comparative device, 
especially if it is illustrated by a time-line. 

Here is a second: 

The earth's orbit is 600,000,000 miles. Represent this 
by a pinhead jV in diameter. Then, 

1. Sun is 1/3400" in diameter. 

2. Earth is 1/340,000" in diameter (invisible under the 
most powerful microscope). 

3. Nearest star is 225 miles away. 

4. Nearest nebula is 30,000 miles away. 

5. Remotest nebulce are 4,000,000 miles away. 

A third: 

The average temperature of the sun's interior is 50,000,000 
degrees (it is probably very much higher). Think of a pinhead 
of matter of this temperature. It would require the energy 
of an engine of 3000 billion horse-power to maintain it. The 

* Possibly much longer than this. 



612 CRAFTSMANSHIP IN MATHEMATICS 

pinhead would emit enough heat to kill anybody 1000 miles 
away. 

This illustration is less successful than the others. There 
is no gradation. A radius of 1000 miles represents an area 
nearly as large as Europe, but to say that a pinhead of matter 
would be so hot as to kill off the whole population of Europe 
does not leave on the mind a sufficiently definite mathematical 
impression. A succession of preparatory stages is desirable. 
To me, however, the temperature in question is utterly un- 
imaginable; I cannot get much beyond the mere arithmetic, 
though I am fairly familiar with the highest temperatures 
that have been produced artificially. 

A fourth: 



For IQQ- of its journey, the light by which we see the 
remotest nebulae travelled towards an earth not yet inhabited 
by man; yet the radius of the universe is 14 times as great 
as the distance of the remotest nebulae. This is obviously a 
better way of bringing home the fact than by giving the length 
of the radius of the universe in miles (2000 million light-years 
= 12,000 trillion miles), though if this number were given 
it would add to the boy's interest to tell him that this 
number is roughly comparable to the number of molecules 
in a single drop of water. 

In dealing with large numbers, it is a sound teaching 
principle (1) to illustrate them by diagrams of some sort, 
(2) to approach them by stages; and it is a simple matter 
to show the boys how these things may best be done. There 
is probably no better plan than that of a succession of distance 
or time lines drawn to gradually diminishing scales. The 
boys will have gleaned the main idea from their history 
lessons. Here is a skeleton history time-line, 55 B.C. to 
A.D. 1931: 

S5&G. 410 1066 J588 1815 1931 



NUMBERS: THEIR UNEXPECTED RELATIONS 613 
Let the line A represent 10 units; B, 100; C, 1000. 



o 
B _ 



c . 


i 

*_ 

10 


2 3 


4 


5 


6 


7 


8 


9 


10 


20 JO 


40 


50 


60 


7O 


80 


90 


100 



100 200 500 -K)0 500 600 700 800 900 1000 

Point out how A has shrunk to yjy of itself in B and to y^o 
of itself in C. The first line might represent a ten-year-old 
boy's 10 birthdays; the second line would then show the 
boy's life history in -^ the length of the first, and C in T ^Q. 
Develop this general idea carefully. 

Now deliberately set a trap that will catch 90 per cent of 
the class: 

" Let us devise a number line the successive parts of 
which shall represent the comparative sizes of really big 
numbers. You know that 10 6 , 10 12 , 10 18 , 10 24 represent, 
respectively, millions, billions, trillions, quadrillions. 



A B 


C 


o 


e 


fc io 6 


IO U 


id 18 


16" 


^ Millions ! 


1 


\ 


1 
| 


'* -r BtUiorts 


1 


1 








1^ Trilli 


^QJ-^5 _ j 


1 


1 


1 






1 



"Is this all-right?" "Yes." " Caught. Surely if 
AC represents a billion it must be a million times as long 
as AB, and AD a million times as long as that. If AB is 
1 inch, AC must be made 16 miles long; AD, 16 million 
miles; and AE 16 billion miles." And so on. 

Books to consult: 

1. Thdwie des Nombres, Desmarest (Hachette). 

2. Theory of Numbers, Peter Barlow (an old book, but still very 
suggestive). 



6U CRAFTSMANSHIP IN MATHEMATICS 

CHAPTER XLIII 

Time and the Calendar 

This subject is essentially mathematical, and it should 

be the business of the mathematical staff to see that the 

following topics are included in their scheme of instruc- 
tion. 

1. Greenwich mean time. How the length of the solar 
day is affected, (a) by the variable movement of the earth 
in its orbit; (j8) by the fact that the axis of the earth is not 
perpendicular to the plane of the orbit. 

2. The modern clock. The astronomical time-keeper is 
a " free " pendulum swinging in a vacuum chamber. How 
its swing is maintained, and how the pendulum of the " slave " 
clock (which does the work of moving the hands round tbf^ 
dial) is made to swing synchronously with it. 

3. Sidereal time; its significance and use. 

4. Summer time; opposition to its adoption. The legal 
definition. 

5. Zone standard times for different countries. How the 
zone of other countries differs from Greenwich time by an 
integral number of hours. Why 5 standard times in U.S.A. 
and Canada and why 3 in Brazil. 

6. The Date or Calendar line. Let the boys examine a 
good Mercator map and discover for themselves how the 
line differs from the 180th meridian. Boys are often 
puzzled about the reason for different days, say Monday 
and Tuesday, on the two sides of this line. Let them 
think the thing out for themselves. The practical difficulty 
of running the line through a group of islands instead of 
round them. 

7. Calendar problems. Successive reforms of the calendar. 
Opposition to further reform religious and social, not scientific. 
Why not 13 months of 28 days each, and one non-calendar 



TIME AND THE CALENDAR 615 

(2 in leap year) day during the year? or some other scheme 
of a more even division than at present? Why even a 
fixed Easter is opposed. How Easter is determined for 
each year. The League of Nations and the Reform of the 
Calendar. 

8. The history of time-measuring. Clepsydras, sand clocks, 
graduated candles, sundials, clocks. 



CHAPTER XLIV 

Mathematical Recreations 

The multitude of problems usually classified under this 
heading may be made a very serious factor of the mathematical 
course. The average boy will face a good deal of drudgery 
if it is a question of solving a puzzle, or seeing his way 
through a trick, or liberating himself from a trap. The 
majority of the so-called mathematical recreations may be 
grouped around definite mathematical principles; if they 
are thus grouped, if the underlying principle of a group is 
thoroughly mastered, and if the members of the group are 
treated as applications of the principle, the work becomes 
as serious as it is interesting. There is no better means of 
giving a boy a permanent interest in mathematics than to 
help him to achieve a mastery of the commoner forms of 
mathematical puzzles and fallacies. A few principal topics 
may be suggested: 

1. Arithmetical puzzles, especially those from mediaeval 
sources. 

2. Geometrical problems and paradoxes. Shunting and 
ferry-boat problems. Paradromic rings. 

3. Chess-board problems. 

4. Unicursal problems. Mazes. 



616 CRAFTSMANSHIP IN MATHEMATICS 

5. Playing-card tricks. } Only partly mathematical, 

6. Cats' cradles. \ but the necessary analy- 

7. Ciphers and cryptographs, j sis is instructive. 

8. Algebraical and geometrical fallacies. 

An isolated problem like Kirkman's school-girls problem is 
also well worth doing, if only for the patient analysis that a 
solution of the problem demands. 

Nearly all the necessary material may be found in the 
late Mr. Rouse Ball's Mathematical Recreations, but the 
literature of the subject is extensive. Every mathematical 
teacher should have on his shelves the works of fidouard 
Lucas; they include everything of interest. The late Henry 
Dudeney's books are also useful. 

I append a manageable figure (it is new) to illustrate the 
principle of Captain Turton's geometrical fallacy, one of 




Fig. 317 

the best I know. Be it remembered that in all cases like this 
the figure is the one thing that matters, if the fallacy is to be 
well concealed. 



MATHEMATICAL RECREATIONS 617 

ABC is an isosceles A of 45, 45, 90. 
Draw CD equal to CA and J. BC. 

Bisect AB in E, join DE and produce to F in CB produced. 
Bisect AF in G and DF in K. Draw GH FA, and KH 
1 FD, meeting in H. Join FH, AH, DH, CH. 

AFGH-AAGH, /. FH-AH; 
AFKH = ADKH, /. FH = DH; 
Hence AH = DH. 

In the As ACH, DCH, 

CA = CD (constr.), 
AH - DH (proved), 
CH is common, 
/. the As are congruent; 

- ZDCH. 



Take away the common angle FCH; 

/. ZACF- ZDCF, 
i.e. 45 = 90, or 1 = 2. 



CHAPTER XLV 

Non- Euclidean Geometry 

What does " non-Euclidean " Mean? 

My experience of the teaching of non-Euclidean geometry 
has been slight (not more than 3 or 4 lessons in all) and 
not very encouraging, though in all cases the teachers were 
certainly competent and the boys (Sixth Form specialists) 
able and well-grounded. Nevertheless, the opinion of many 
prominent mathematical teachers is that boys ought to 



618 CRAFTSMANSHIP IN MATHEMATICS 

know something about the subject. Personally I think it is 
too difficult, and is best taken at the University later. 

It is essential for a teacher who decides to include the 
geometry in his Sixth Form course, first to familiarize himself 
both with the whole subject and with its implications. Here 
is a possible first course of reading: 

1. The controversies of the last hundred years concerning 

Euclid's parallel postulate. 

2. Some such book as Hilbert's Foundations of Geometry. 

3. Part II of Poincare's Science and Hypothesis. 

4. Mr. Fletcher's article in No. 163 of the Mathematical 

Gazette, viz. " A method of studying non-Euclidean 
geometry ". 

Captain Elliott writes a suggestive short article on " Practical 
non-Euclidean Geometry " in No. 177 of the Gazette. 

At the beginning of the nineteenth century, almost 
simultaneously, Lobatscheffski, a Russian, and Bolyai, a 
Hungarian, showed irrefutably that a proof of the parallel 
postulate is impossible. It will be remembered that Euclid 
himself seemed to recognize a difference in the degree of 
conviction carried to the mind by his statement concerning 
parallels, compared with that of his other fundamental assump- 
tions; and he called the statement a postulate rather than 
an axiom. It is incorrect to include the statement as his 12th 
axiom, as is commonly done. 

Lobatscheffski assumed that through a point an infinite 
number of parallels may be drawn to a given straight line, 
but he retained all the other basic assumptions of Euclid. 
On these foundations he built up a series of theorems which 
are perfectly self-consistent and non-contradictory; the 
geometry is as impeccable in its logic as Euclid's. The 
theorems are, however, at first sight disconcerting; for 
instance, (1) the sum of the angles of a triangle is always less 
than two right angles, and the difference between that sum 
and two right angles is proportional to the area of the triangle; 
(2) it is impossible to construct a figure similar to a given 



NON-EUCLIDEAN GEOMETRY 619 

figure but of different dimensions. Lobatscheff ski's pro- 
positions have little or no relation to those of Euclid, but 
they are none the less logically interconnected. Let the 
reader try to reconstruct, say, the first 32 propositions of 
Euclid, Book I, on the assumption that the parallel postulate 
(the " 12th axiom ") is untenable; he will probably be more 
than a little surprised. 

Riemann, a German mathematician, likewise rejected 
Euclid's parallel postulate. He also rejected the axiom that 
only one line can pass through two points. Otherwise he 
accepted Euclid's assumptions. The system of geometry 
which he then built up does not differ essentially from spherical 
geometry. On a sphere, through two given points, we can 
in general draw only one great circle, the arc of which between 
the two points therefore represents the shortest distance, 
and hence the straightest line between them. But there is 
one exception; if the two given points are at the ends of 
a diameter, an infinite number of great circles can be drawn 
through them. In the same way, in Riemann's geometry, 
through two points only one straight line can in general 
be drawn, but there are exceptional cases in which through 
two points an infinite number of straight lines can be drawn. 
Riemann's " space " is finite though unbounded, just as 
the surface of a sphere is finite though unbounded. 

Thus Lobatscheffski's and Riemann's geometries, though 
both non-Euclidean, were, in a measure, opposed to each 
other. 

In Euclid's geometry, the angle-sum of a triangle is two 
right angles, in Lobatscheffski's less than two right angles, 
in Riemann's greater than two right angles. 

In Euclid's geometry, the number of parallel lines that 
can be drawn through a given point to a given line is one; 
in Lobatscheffski's, an infinite number; in Riemann's, none. 

Euclidean geometry (which includes all school geometries) 
retains the parallel postulate; non-Euclidean geometries are 
those which reject the postulate. 



620 CRAFTSMANSHIP IN MATHEMATICS 

Which Geometry is True? 

, Mr. Fletcher asks the question, which geometry is true? 
and answers it by saying that they are all true, though the 
whole question turns on the nature of a straight line. The 
straight line, being elementary and fundamental, cannot, 
however, be " defined ", for there is nothing simpler in 
terms of which it can be expressed. Thus we are driven 
to an indirect definition by axioms. But inasmuch as the 
axioms of the three geometries differ, it is obvious that 
they define different things. But by " things " we mean 
not material objects but ideas suggested by them. All three 
geometries are true but only as applying to the " things " 
known or unknown to which they refer. 

But when it comes to the application of geometry to the 
facts of the external world, the question is, as Mr. Fletcher 
points out, different. " Now we have to ask: which of these 
absolute pure sciences applies the most conveniently or the 
most exactly to the facts with which we are dealing? In large 
scale work, where alone the differences in the results of 
the geometries are large enough to be apparent, we are dealing 
chiefly with the form of a ray of light, or the line of action 
of gravitation. It is easy to see that these forms may differ 
according to circumstances; that while, as seems now to be 
probable, Euclid's geometry may be applicable with all 
necessary exactness to those rays * at an infinite distance ' 
from gravitating matter, Lobatscheff ski's, or more probably 
Riemann's, may afford a better tool for dealing with them 
in the neighbourhood of such matter." 

Since in the material world in which we live, Euclid's 
parallel postulate seems to be satisfied, Euclid's geometry 
will continue to be the geometry of practical life and hence 
of our schools. 

" The essential requisite for clear thinking on the* subject," 
Mr. Fletcher says, " is the maintenance of the distinction 
between the pure science and the applied. The * things ' 
with which the former deals are ideas, abstractions; it can 



NON-EUCLIDEAN GEOMETRY 621 

only proceed from axioms, but on that basis its results 
are absolute. The latter deals with * facts ', with * external 
things '; its basis is experimental, and its results approximate. " 
Impress upon Sixth Form specialists that in our ordinary 
geometry we always argue as if we were living on a plane, 
whereas really we are living on a sphere. The surface of the 
very table we write on is, strictly, part of the surface of a 
sphere of about 4000 miles radius. Practically, it is a plane, 
of course, but if we build up a theoretical system (as Euclid 
did) on the assumption that we are dealing with actual planes 
instead of with parts of a spherical surface, how can our 
system be free from possible fallacy? And if we apply that 
system to the measurement of stellar distances, how can we 
logically assume that it is strictly applicable? 

Relativity 

It is not a difficult matter to give boys a clear understanding 
of the special theory * of Relativity, but the general theory 
is much too difficult for them. Even so, it is possible to 
give them one or two comparatively elementary lessons on the 
general theory, to enable them to see that the final acceleration- 
difference in Newton's and Einstein's gravitation formulae, 
unlike as these formulae are in appearance, is almost insigni- 
ficant. Do not confuse " space-time " with hyperspace, a 
totally different thing. Space-time is merely a mathematical 
abstraction devised to meet the indispensable need of consider- 
ing time and three-dimensional space together. The greater 
part of Professor Nunn's admirably written book on Relativity 
can be understood by Sixth Form specialists, as I know from 
experience. 

Hyperspace 

When we come to the question of hyperspace, we are in 
a region of difficulty too great for all but the very exceptional 

* The Relativity of Simultaneity is apt to be a little puzzling. For suggestions 
see Science Teaching, pp. 357-72. 



622 CRAFTSMANSHIP IN MATHEMATICS 

boy. Even some professional mathematicians still have deeply- 
rooted prejudices against N-dimensional space. The crudest 
form of prejudice is what may be called the " common- 
sense " opinion that as space cannot have more than three 
dimensions, any consideration of hyper-space is obviously 
nonsense. When Einstein announced his general theory, 
a distinguished Oxford philosopher wrote an indignant 
letter to The Times, pointing out (amongst other things) that 
inasmuch as Aristotle himself had pronounced space to be 
just long and broad and deep, in other words three-dimen- 
sional, there was nothing more to be said: Einstein's irrever- 
ence was almost unpardonable. Evidently the writer of that 
letter was under a complete misapprehension as to the nature 
of Relativity. Einstein's space is not four-dimensional but 
three-dimensional, though cosmically it is probably not quite 
homaloidal but slightly curved. Einstein's fourth dimension 
is time, not space. The same writer also probably misunder- 
stood the nature of geometry as a science. Geometry certainly 
did start as a form of " earth-measuring ", but even in the 
time of the ancient Greeks it had developed into a semi- 
abstract science, to be deduced from a limited number of 
axioms and definitions. For more than 2000 years after 
Euclid, it was supposed that axioms were self-evident truths 
about the real world. Only one axiom, that concerning 
parallels, fell short of the high standard of the others: it 
was not self-evident. Attempts to prove it all failed, and 
at last it was realized that a logical system of geometry could 
be constructed by starting with the denial of the axiom, or 
postulate as it ought to be called. 

Students of Relativity need not concern themselves much 
with hyperspace, but it must not be thought that hyper- 
geometry can have no application to the geometry of the real 
world or to physics. Beginners in wave mechanics naturally 
assume that the three dimensions required in Schrodinger's 
theory of the motion of a single particle are the three 
dimensions of ordinary space, but as soon as we come to two 
particles six dimensions are required. Many problems in 



NON-EUCLIDEAN GEOMETRY 623 

thermodynamics require a number of dimensions exceeding 
three, though perhaps " degrees of freedom " rather than 
" dimensions " is a term more acceptable to some people. 

Books to consult: 

1. Einstein's Nottingham University Lecture of June 6, 1930. 
(See Nature for June 14.) 

2. Professor Forsyth's address, Dimensions in Geometry y to the 
Mathematical Association. (See Gazette, 212.) This address is 
most illuminating. 

3. Professor Sommerville's Geometry of N Dimensions. 

4. Professor Baker's Principles of Geometry, Vol. IV. 

(The last two books should be read by teachers of mathematics, 
but, of course, for advancing their own knowledge, not for actual 
teaching purposes.) 



CHAPTER XLVI 

The Philosophy of Mathematics 

Mathematical teachers will be well advised to admit 
that the philosophical foundations of mathematics is a subject 
which is outside the limits of Sixth Form work, save in the 
case of very exceptional boys. There are, however, a few 
points of a sufficiently simple character that can be included, 
if only in order that boys may, before leaving school, lose 
some of the " cocksureness " that early mathematical success 
so often excites, and learn that mathematical truth is, after 
all, something that is still far from being absolute, something 
that is still relative. 

Mathematical and other Reasoning 

One or two formal lessons on the general nature of 
reasoning, deductive and inductive, are advisable. This does 
not mean that time should be spent on formal logic, except 
in so far as is necessary for a clear understanding of the 



624 CRAFTSMANSHIP IN MATHEMATICS 

syllogism, and that is necessary. The mere setting-out of 
mathematical truth, as distinguished from the search and 
discovery of it, is essentially syllogistic, synthetic, and deduc- 
tive in character. But mathematical reasoning is not deductive; 
it is above all things analytical, inductive. 

Let the boys understand clearly that the elementary 
scheme of syllogistic reasoning that at one time passed for 
" logic ", the formal deductive logic of the last 2000 years, 
does not really represent our ordinary modes of reasoning, 
but is rather a scheme by which we try to show other people 
how our conclusions follow from our premisses. The arranging 
of a string of syllogisms, as syllogisms, presents no serious 
difficulty; from accepted premisses, a logical conclusion 
follows, with almost mechanical precision. Professor Jevons 
actually invented a logical machine, almost as simple as a 
penny-in-the-slot-machine, which made clear that syllogistic 
reasoning was at bottom mechanical. In reasoning the real 
difficulty is concerned with the premisses that compose 
the syllogism, not with the syllogism itself. Can we certify 
that the premisses are true, and do we all agree about the 
exact significance of the terms we use? This is the trouble, 
and the only real trouble, involved in ordinary reksoning. 
To follow out the sequence of syllogisms in a Euclidean 
proposition is child's play. 

Opposite is a scheme showing Euclid's chain of reasoning 
for proving I, 32. The proposition and all the propositions 
on which it depends are easy enough to follow up, and may 
(with certain exceptions) be accepted. But when we come 
to the axioms which form the ultimate premisses of the pro- 
positions, and examine them carefully, we begin to feel 
doubt and difficulty. In short, we have reached the point 
where the work of serious reasoning begins. 

This is the first thing for boys to bear in mind: that 
although mathematical demonstrations have every appearance 
of being mere chains of syllogisms, we may not infer that 
mathematical reasoning is deductive. The very contrary is 
really the case. Mathematical reasoning is above all things 



THE PHILOSOPHY OF MATHEMATICS 
1 
2 



4 

I 



625 



5 

7 
8 



22 



23 



9 
10 



11 
13 
15 



1(3 
27 



29 



31 



32 

analytical, inductive; it cannot be reduced to the rules of 
deductive logic. We dress the results up syllogistically 
merely for exhibition purposes. 

Impress upon the boys that in all reasoning beginners 
tend to believe that the premisses are true if acceptable 
consequences seem to follow from them, and the longer the 
chain of apparently sound, intermediate links, the less sus- 
picious they become of any weakness in the first link; but 
that conclusions are quite worthless unless the premisses on 
which they first depend are unassailable. 

It was John Stuart Mill who in the middle of the last 
century first shook our faith in Aristotelian logic. Mill's own 
position has since been shaken, but it was he who first gave 
to the subject its proper outlook. 

(E291) 41 



626 CRAFTSMANSHIP IN MATHEMATICS 

The " new " logic, the logic especially of the last thirty 
years, insists upon this: that verbal explanations of meaning, 
so long as they remain merely verbal, are futile. Merely 
to " infer " one proposition from another, and to go on doing 
this for ever, gives us nothing but unexplained " propositions " 
at every step of the process. The old logic never even became 
aware of the fatal confusion between assertion and sentence 
that is covered by the word " proposition ". According 
to it, if we get one proposition from another or others, we 
have arrived at the end of a process of inference and have 
obtained a " conclusion ". If " all men are feathered animals " 
happened to be one of the premisses, the old logic did not 
question that premiss at all, but proceeded to draw " con- 
clusions "; the truth or falsehood of the premisses was none 
of its business. Its conclusions were thus often meaningless 
and misleading; sham, not real. The new logic, on the 
other hand, pays first attention to the premisses, knowing 
that the 3ubsequent process of inference is a process relatively 
elementary. 

We must not allow our admiration of the Greeks to blind 
us to their limitations and their failures. Greek mathematics 
was great; modern mathematics is greater. In certain* regions 
the Greeks failed where we feel they ought to have succeeded. 
We feel, for example, that they ought to have anticipated 
Descartes; and we feel that, with their power of generalization 
and their love of philosophy, they ought to have given some 
coherent account of the foundations of geometry. The 
early pages of Euclid are definitely unsatisfactory. 

Apparently the Greeks never realized that the foundations 
of geometry were necessarily abstract. Even Euclid himself 
could only look at space vaguely, and give some sort of popular 
description of what he thought he saw there. And "to the 
Greeks generally, geometry was always the " science of space" 
of the physical space of experience in which we live. It is 
quite impossible to base a coherent geometry on such a 
foundation; the superstructure may be magnificent, but it 
is always likely to overturn because of the instability of its 



THE PHILOSOPHY OF MATHEMATICS 627 

foundations. That the theorems of geometry are not affected 
by earthquakes, that the Greeks could understand, but they 
could not, or at least did not, understand that geometry has 
nothing to do with physical space; that its " space " is its 
own creation; and that it is simply the statement of the logical 
relations between objects, defined by these relations alone. 

Non-mathematicians still commonly suppose that the 
early pages of Euclid, his " axioms " and " postulates ", are 
profound and never to be questioned. There is, in point of 
fact, not one of these axioms or postulates, the parallel 
postulate alone excepted, which has anything but an historical 
interest, or which embodies any permanent contribution to 
science. Here the Greeks failed altogether. 

Here is Mr. Bertrand Russell's opinion of Euclid. 
" When Euclid is attacked for his verbosity or his obscurity 
or his pedantry, it has been customary to defend him on 
the ground that his logical excellence is transcendent, and 
affords an invaluable training to the youthful powers of 
reasoning. But at a close inspection, this claim vanishes. 
His definitions do not always define, his axioms are not always 
indemonstrable, his demonstrations require many axioms 
of which he is quite unconscious. 

" The first proposition assumes that the circles used in 
the construction intersect an assumption not noticed by 
Euclid because of his dangerous habit of using a figure. The 
fourth proposition is a tissue of nonsense. Superposition 
is a logically worthless device; for if our triangles are spatial, 
there is a logical contradiction in the notion of using them; 
if they are material, they cannot be perfectly rigid, and when 
superposed they are certain to be slightly deformed from 
the shape they had before. The sixth proposition requires 
an axiom for proving that if D be in AB (the side of the 
isosceles triangle ABC), and BD is < BC, the triangle, DBC 
is < the triangle ABC. The seventh proposition is so thor- 
oughly fallacious that Euclid would have done better not 
to attempt a proof. I, 8 involves the same fallacy as I, 4. In 
I, 9 we require the equality of all right angles, which is not 



628 CRAFTSMANSHIP IN MATHEMATICS 

a true axiom since it is demonstrable. I, 12 involves the 
assumption that a circle meets a line in two points or in none, 
which has not in any way been demonstrated. I, 26 involves 
the same fallacy as I, 4 and I, 8. Many more criticisms might 
be passed on Euclid's methods, and on his conception of 
geometry; but the above definite fallacies seem sufficient to 
show that the value of his work as a masterpiece of logic 
has been very grossly exaggerated." 

Still, the Greeks did conceive the universe as a cosmos 
subject to rule; they did recognize that the universe is, at 
bottom, a mathematical affair. It was not until 2000 years 
later that the foundations of their science were carefully 
examined and found to be lacking in any sort of solidity 
or permanence. 

Axioms, Postulates, Definitions 

Every conclusion rests on premisses. These premisses 
are either self-evident and require no demonstration, or they 
can be established only by demonstration from other proofs. 
Since we cannot thus proceed in this latter fashion ad mfinitum, 
geometry must be founded on a certain number of un- 
demonstrable propositions. 

To these undemonstrable propositions, some of the 
greatest mathematicians of the last fifty years have devoted 
very serious attention. That there is still divergence of 
opinion shows how difficult the subject really is. The several 
books which are the separate or joint productions of Mr. 
Russell and Professor Whitehead will be familiar to most 
readers, and the books of equally eminent French and German 
mathematicians will also be familiar to some. Such a mass 
of authoritative literature will convince any non-expert that 
the subject is a very thorny one. 

The standard of logical rigour in mathematics is now 
greater than it has ever been. It is, however, still quite 
permissible, in teaching, to make use of small boys' in- 
tuitions to use them, in some measure, as a reinforcing 



THE PHILOSOPHY OF MATHEMATICS 629 

basis when establishing elementary principles from data 
derived from the boys' experience or from special experi- 
ment. For later serious work, geometrical intuitions are 
not sufficiently trustworthy. One well-known authority 
describes intuitions as " a mere mass of unanalysed prejudice ". 
Certainly boys' intuitions are necessarily often crude, often 
meaningless, just shots in the dark. We shall return to the 
question of intuition in the next chapter. 

What are geometrical axioms? Kant said they were 
synthetic a priori intuitions. But in that case they would 
be imposed upon us with such a force that we could not 
conceive their contraries, and this we now know we can do. 

Are they, then, experimental truths? We do not make 
experiments on ideal lines or ideal circles; we can make 
them only as material objects. On what, therefore, would 
experiment serving as a foundation for geometry be based? 
The answer is simple. Since we constantly reason as if 
geometrical figures behaved like solids, geometry can borrow 
from experiments only the properties of those bodies. But 
this involves an insurmountable difficulty, for if geometry 
were an experimental science, it would not be an exact 
science; it would be subjected to continual revision. Indeed 
no rigorously invariable solid exists. 

Thus the axioms are neither synthetic a priori intuitions 
nor experimental facts; they are merely conventions. 

Our choice amongst all possible conventions is guided 
by experimental fact, but it remains free, and it is limited 
only by the necessity of avoiding every contradiction. Thus 
different geometries are possible according to our initial 
choice of conventions. One of these geometries is not more 
" true " than another (cf. the last chapter); it can only be 
more convenient. Euclidean geometry is the most convenient 
because (1) it is the simplest; (2) it sufficiently agrees with 
the properties of natural solids, those bodies which we can 
compare and measure by means of our senses. 

Henrici (no mean geometer) called Euclid's definitions 
" axioms in disguise ". Poincar6 (a much abler mathematician, 



630 CRAFTSMANSHIP IN MATHEMATICS 

perhaps the greatest of the last fifty years) called Euclid's 
axioms " definitions in disguise ". As to Euclid's postulates, 
some writers have seen in them merely statements which 
limit the use of instruments in geometrical construction to 
ruler and compasses, but this is a mere side issue, and has 
nothing whatever to do with the nature of geometrical reason- 
ing. 

Euclid himself grouped the 10th, llth, and 12th axioms 
together with the three postulates, in one class, under the 
name arn?/xara; the arrangement found in the modern editions 
of his Elements is the work of his successors, the ground of 
the alteration being, " the distinction between postulates and 
axioms, which has become the familiar one, is that they 
are the indemonstrable principles of construction and demon- 
stration, respectively ". This distinction is not accepted by 
the modern school of geometers and should be ignored. 
Views on the question are, however, discordant. 

We referred in an earlier chapter to the desirability of 
teaching boys carefully how to formulate their own de- 
finitions. The great purpose of a definition is the precise 
discrimination, yes or no, of actual cases. The definition 
must leave no doubt at all as to the identity of the thing 
defined. With beginners there is bound to be superfluity 
of statement, but refinement will come as the years go on. 
Never provide pupils with definitions ready made. 

I find that very few boys are able to grasp the significance 
of the argument, pro and con, as to whether axioms (if, as 
they seem to be, they are definitions in disguise) are a priori 
intuitions, or experimental truths, or geometrical abstractions, 
or whether they define conventions. The arguments are for 
maturer minds. But it is easy to bring a boy to see that we 
are all liable on occasions to admit as self-evident propositions 
which we subsequently recognize not to be so. 

Ask a class (a Fourth Form will do) to consider the axiom, 
" two straight lines cannot enclose a space ". Accept their 
usual definition of a straight line, viz. " the shortest distance 
between two points ", as determined by a stretched string. 



THE PHILOSOPHY OF MATHEMATICS 631 

Then obtain from them the admission that the straightest 
the most direct line (the geodesic) that can be drawn 
between two points on the surface of a sphere may be de- 
termined in the same way; then show that this line is necessarily 
part of a great circle. Next, 1 show that any two such lines 
on the surface of a sphere, being parts of great circles, intersect 
in two points; hence, at all events on the surface of a sphere, 
any two " straight " lines must enclose a space. 

But we live on the surface of the sphere, and any surface 
we call plane, no matter how small, must be part of a spherical 
surface, and therefore any two " straight " lines we draw, 
being parts of great circles, must meet, at two points 180 
apart. Even if the two lines we draw on the paper are what 
we call " parallel ", the same thing applies. 

Point out that this is not mere theory, it is prosaic fact, 
for (1) the earth is known to be a sphere and therefore all 
our so-called planes are parts of a spherical surface; and 
(2) canal engineers actually have to allow 8" in the mile for 
sphericity; if they made the bottom of the canal absolutely 
" level' \ they would be cutting a channel which would eventu- 
ally emerge at the spherical surface. If 3 poles of equal length 
be set up at equal intervals on a perfectly " level " stretch 
of land, say along the straight six miles of the Bedford level 
between Witney Bridge and Welsh's Dam, and a telescopic 
sight be taken from the top of the first to the top of the 
third, the line of sight will be seen to pass 5 or 6 feet below 
the top of the second. If from one end of the Corinth canal 
we look through the clear Greek air along the canal surface, 
the earth's curvature is readily seen-, and the amount of 
curvature is easily measured by taking from one end of the 
canal a tangential telescopic sight to meet some object, say 
the hull of a ship, at the other end.* 

In a Sixth Form the argument can be carried farther, 

* There are no locks on the canal, and two of its four miles of length run 
through a cutting with an average depth of 200 feet. Tested at any point of its 
length, the canal surface is found to be perfectly "level", yet the water half-way 
along is demonstrably several feet " higher " than at either end. 



632 CRAFTSMANSHIP IN MATHEMATICS 

and conviction carried home. At all events, enough can be 
done (with other axioms as well as this) to make boys critical, 
in the future, of principles that claim to be fundamental. 
An elementary lesson on " infinity " may serve to lead up 
to an analysis of such an axiom as " the whole is greater 
than its part ". On two or three occasions I have known 
useful Sixth Form discussions take place after the boys had 
read the more elementary parts of Mr. B. Rus*sell's chapters 
on Infinity: see, for instance, pp. 179-82 of his Knowledge 
of the External World. Some notion of the mathematical 
significance of the term infinity should be given to Sixth 
Form boys; but the difficulties are real. Mr. Russell is our 
ablest expositor of these difficulties. His critical views are 
drastically destructive; his constructive views are not accepted 
by all mathematicians. 

Mathematical Proof 

We have all heard a small boy, when engaged in a lively 
argument with another small boy, say suddenly, " You can't 
prove it." What does he really want when he thus calls for 
a certificate of proof? It is very hard to say. Verification by 
demonstration? Authority? 

Strictly, the proof of a proposition is its directly logical 
syllogistic derivation from other propositions which we 
know to be certain and necessary, and ultimately derivative, 
therefore, from definitions and axioms. To that extent every 
deduction from definitions and axioms is also the proof 
of the conclusion reached by it. 

When a distinction is made between proof and deduction, 
the proof is regarded as the problem of deciding as to the 
truth of an hypothesis, of confirming it or refuting it. The 
proof of an affirmative is the refutation of the negative; 
and vice versa. 

The mathematical forms of statement that have been 
devised to record the facts of a proof include the explicit 
mention of all the considerations needed to justify it against 



THE PHILOSOPHY OF MATHEMATICS 633 

any attack. But this record is made after the proof has been 
achieved, and in setting out the proofs we keep out of the 
record all our unsuccessful attempts, all our " scrap-paper " 
work, and include just those few links in the main chain that 
are well and truly forged. 

Ultimately we go back to our definitions and axioms, 
and it is here where we are really so susceptible to attack. 
We never seem to be able to make ourselves armour-proof. 
Mr. Bertrand Russell, or somebody like him, will come 
along and inevitably find a joint where he can inflict a nasty 
wound. " Proof " is purely a question of degree. It is hope- 
less to attempt to find a means of dissolving mathematical 
error completely, and of exhibiting Truth in a white light, 
unassailable. The most we can do with boys is to train them 
both to be always on the look-out for mathematical shams 
and to hunt these down relentlessly. 

It is fatuous to make Third or Fourth Form boys write 
out the general " proof " of, say, the division process of 
finding the algebraic H.C.F. What does the " proof " signify 
to them? So it is with " proofs " all the way up the school. 
When a Fifth Form boy has " proved " the binomial theorem 
ask him what it is all about; closely cross-examine him and 
show him how the term " prove " has been improperly used. 
Let him establish the binomial theorem; let him show that 
for a fractional index the theorem takes the same form as 
for an integral index, but do not let him pretend to " prove " 
either. Beware of using the term proof in connexion with 
mathematical induction. Even the sometimes substituted 
phrase " proof by repetition " is open to criticism. 

44 Pure " Mathematics 

If a strictly logical treatment of mathematics implies, as 
some present-day mathematicians and others contend, a 
strictly abstract treatment, the objects with which mathematics 
deals are just symbols, devoid of content except such as is 
implied in the assumptions concerning them. This abstract 



634 CRAFTSMANSHIP IN MATHEMATICS 

symbolism constitutes what is sometimes called " pure " 
mathematics, everything else being, strictly, " applied " 
mathematics, since it deals with concrete applications of an 
abstract science. Thus all the ideas of pure mathematics 
can (so it has been seriously contended) be defined in terms 
which are not strictly mathematical at all, but are involved 
in complicated thought of any description. If this be true, 
all the propositions of mathematics might be deduced from 
propositions of formal logic. 

The distinction is not the same as the distinction of thirty 
or forty years ago. Then, " pure " mathematics included all 
ordinary work in algebra, geometry, and the calculus;/ applied 
mathematics included such subjects as mechanics, surveying, 
and astronomy. Pure mathematics was the mathematics of 
the blackboard; applied mathematics was supposed to be 
the mathematics of the laboratory, but too often experiment 
played no part at all. The " pure " mathematician was a 
very exclusive sort of person, rather despising those who 
did the weighing and measuring necessary hacks they 
admitted, but hacks all the same. 

Nowadays " pure " mathematics tends to shrink into a 
smaller compass. Formerly, the laboratory experiment with, 
e.g., Fletcher's trolley came (as it still comes) within the 
ambit of applied work, but all subsequent considerations of 
the curve produced belonged to algebra and trigonometry, 
and therefore constituted " pure " work. But not so now. 
The curve itself is now recognized as a thing of ink or chalk, 
and is therefore material; it is not really a geometrical curve, 
it is a black thing or a white thing that we make, and is only 
a crude representation of the true curve which, if we are 
" pure " mathematicians, it behoves us to consider. 

Thus " pure " mathematics tends to become a new* 
subject, a subject in the border region between the mathematics 
that ordinary people learn, and philosophy. The subject is 
a very serious one, a subject within the realm of " pure " 
or abstract thought; but it is not a subject for schools. It 
was a prominent Church paper that, a few years ago, be- 



THE PHILOSOPHY OF MATHEMATICS 635 

lauded the " pure " mathematician because he was a " good " 
man; the " applied " mathematician was necessarily led by 
his " impure " work to free thinking and infidelity! The 
small circle of eminent philosophical mathematicians whom 
we recognize as authorities on abstract mathematics must 
be proud of their testimonial. But the great majority 
teachers and all others who are engaged in ordinary worka- 
day mathematics, merely " applying " the basic principles 
laid down by the few should ponder over the fate that 
is said to await all infidels! 

Is it not a little just a little absurd to pretend that 
we teach " pure " mathematics in schools. The work we 
do is all applied work, the different forms of which are all 
a question of degree. As we go up the school, the concrete 
work receives a gradually deepening tinge of abstractness, 
but even in the Sixth the work is never more than partially 
abstract. To claim that the Upper Form geometry we teach 
is more " refined ", is " purer ", or is " more intellectual " 
than mechanics or astronomy is merely to provoke ribaldry. 

It is sometimes said that Newton was a " pure " mathe- 
matician. But was he? He spent his life in rounding off the 
work of the astronomers from the time of the ancient Greeks 
and Egyptians to the time of his predecessor Galileo. Even 
the new mathematical weapon (the calculus) which he forged 
was forged for the purpose of pushing ahead with his investiga- 
tions among real things. But he was certainly also a philosopher 
if by this term we mean a thinker who looks to his foundations. 
His researches did not, however, take him very far into 
purely abstract mathematics. He was too busy with such 
problems as that of showing that a falling stone and a falling 
moon are subject to the same law. 

Though men could not deny the tremendous success of 
Newton's system of mechanics, though Laplace acclaimed it 
as final, yet there remained questionings. One distinguished 
critic after another felt doubts about his absolute space and 
time, and Einstein, setting out to satisfy them, not only did 
so, but in the doing he evolved a theory that included not 



636 CRAFTSMANSHIP IN MATHEMATICS 

only all that Newton had done but those other points which 
Newton's theory could not be made to include. 

Mathematics, like all other subjects, has now to take 
its turn under the microscope and reveal to the world any 
weaknesses there may be in its foundations. But this is not 
work for schoolboys. To boys the main objects of mathe- 
matical study must continue to be real things, even if those 
real things are only figures produced by ruler and compasses. 
Mr. Russell criticized Euclid for drawing figures, because 
those very figures were partly responsible for preventing 
Euclid from building up the flawless system he aimed at. 
Schoolboys, working on a lower plane, would, without figures, 
be helpless. 

Relative Values 

In one respect at least, mathematics seems to be a subject 
quite unlike other subjects. Its discoveries are permanent. 
The theorem of Pythagoras for instance is as valid to-day as 
2000 years ago. The majority of the mathematical truths 
we now possess we owe to the intellectual toil of many 
centuries, and a student who desires to become a mathe- 
matician must go over most of the old ground before he 
can hope to embark on serious research. To the uninitiated 
it is impossible to make mathematical truths clear. The 
great theorems and the great results of mathematics cannot 
be served up as a popular dish, and this inaccessibility of 
the subject tends to make it rather odious to those whose 
early grounding was of little account. 

Although mathematics does not lead to results which are 
absolutely certain, the results are incomparably more trust- 
worthy than those of any other branch of science. Still, if 
mathematics stands aloof, if it is not turned to practical account 
in other branches of science, it remains a useless accumulation 
of capital, almost an accumulation of lumber. 

Can it be maintained that, as an intellect-developing in- 
strument, mathematics ranks first amongst the different 
subjects of instruction, or even " first among equals "? 



THE PHILOSOPHY OF MATHEMATICS 637 

Mathematical reasoning is in some respects simpler than 
scientific reasoning; the data are clearer. It is simpler than 
linguistic reasoning; there are no probabilities to weigh. It 
is simpler than historical reasoning; there are no difficult 
human factors to consider. But in one respect mathematical 
reasoning is the most difficult of all, and this is because of 
the inherent difficulties of mathematical analysis, whether the 
analysis is the Third Form analysis of the data of a simple 
geometrical rider, or whether it is the more serious analysis 
in Sixth Form work. Relatively speaking, there is no other 
difficulty. The analysis once effected, the rest is plain 
sailing. 

I attach very great value to mathematical instruction, but 
I deny that the virtue of the instruction lies in anything of 
the nature of super-certitude. 

Picture -making by Physicists: the Dangers 

It has been said that, from the broad philosophical stand- 
point, the outstanding advances in the physics of the last 
20 years have been the theory of Relativity, the theory of 
Quanta, the theory of Wave-mechanics, and the dissection 
of the atom. But it would be more correct to say that it is 
the surrender of our rather aggressive certitude about the 
nature of things, and our recognition that we are still ignorant 
of the nature of ultimate reality. We now know that our 
pictures were all wrong. How faithful we were to the aether 
as a quivering jelly of inconceivable density! How we loved 
to tie knots in aetherial vortex- rings! 

Certain seaside resorts are on a cliff, with an upper 
esplanade brilliantly illuminated at night, and, 10 or 15 feet 
below, an unlighted walk protected by a low wall. The 
shadows of passing people and vehicles on the upper esplanade 
are cast upon this low wall, and may be watched by a person 
seated on the lower level in the dark. Imagine such a person 
to have been entirely cut off from human kind since his 
early childhood, to be fastened to his seat permanently in 



638 CRAFTSMANSHIP IN MATHEMATICS 

the dark, and to see nothing but the shadows in front of him. 
We may consider him endowed with powers of reasoning 
and with some amount of mathematical power. 

We can imagine him observing the shapes, sizes, move- 
ments, and velocities of the shadows, gradually sorting out 
resemblances and differences, and eventually establishing a 
number of equations embodying the whole of his sense 
data. These equations would be strictly representative of 
reality as he knew it. But suppose he now began to speculate, 
and to attempt to infer from his equations the nature and 
properties of the original things, animate and inanimate, 
that had cast the shadows on the wall. Would not the rapidly 
moving motor-car be given pride of place, and would not 
the slow-moving human being be looked upon as of secondary 
importance? Would not all his conjured-up mental pictures 
unfailingly be a mere travesty of reality? In his allegory 
of the cave, Plato warned us, more than 2000 years ago, of 
fallacies of this kind. 

Physicists are learning that the greater part of their 
observations are not observations of reality but of the shadows 
of reality. When they invent an atomic gymnasium and a 
system of electronic gymnastics, they know well that they are 
just speculating wildly. On the other hand they know that 
the mathematical formula they have established is, though 
uninterpretable, in some way representative of reality. 

Einstein's formula for gravitation is universally accepted. 
His cosmology is not accepted. No cosmology can be, 
for it is necessarily hypothetical, speculative, fanciful. 

Teach the boy that the physicist as a research worker 
and mathematician is a man to be respected, but that while 
we may admire his pretty pictures, we are quite certain that 
none of these will ever make Old Masters. 

Books to consult: 

1. Professor Whitehead's books. 

2. Mr. Bertrand Russell's books. 

3. Mathematical Education, Carson. 



THE PHILOSOPHY OF MATHEMATICS 639 

4. Mysticism in Modern Mathematics, Hastings Berkeley. 

5. Science and Hypothesis, Poincare. 

6. Les Stapes de la philosophie mathdmatique , Brunschvicg 

7. De la Certitude logique, Milhaud. 

8 The Human Worth of Rigorous Thinking , Keyser. 



CHAPTER XLVII 

Native Genius and Trained Capacity 

Russell versus Poincar6 

Mathematical philosophers, like the philosophers of other 
schools, naturally have greater faith in their own systems 
than in the systems of their rivals. Over one point in particular 
they are hopelessly at variance, namely, as to the respective 
roles that logic and intuition play in the origin and development 
of mathematical ideas. 

It was Aristotle who worked out the principles of de- 
ductive logic, and his scheme was universally accepted 
almost down to the close of the Victorian era. In the middle 
of the last century, George Boole, a distinguished mathe- 
matician, pointed out how deductive logic might be com- 
pletely symbolized in algebraic fashion. Given any pro- 
positions involving any number of terms, Boole showed 
how, by a purely symbolic treatment of the premisses, logical 
conclusions might infallibly be drawn. 

At the beginning of the present century (in 1901), Mr. 
Bertrand Russell said,* " Pure mathematics was discovered 
by Boole. His work was concerned with formal logic, and 
this is the same thing as mathematics "; and again:-)- " The 
fact that all mathematics is symbolic logic is one of the 
greatest discoveries of our age, and the remainder of the 

In the International Monthly. f Principles of Mathematics. 



640 CRAFTSMANSHIP IN MATHEMATICS 

principles of mathematics consists in the analysis of symbolic 
logic itself. " In their Principia Mathematica> the aim of 
Mr. Russell and Professor Whitehead is to deduce the whole 
of mathematics from the undefined logical constants set forth 
in the beginning. And in Signor Peano's Formulario, the 
different branches of mathematics are " reduced to their 
foundations and subsequent logical order ". Moreover, in 
his work Les Principes des MatMmatiques, M. Couturat 
expresses the opinion that the works of Russell and Peano 
have definitely shown not only that there is no such thing 
as an a priori synthetic judgment (i.e. a judgment that 
cannot be demonstrated analytically or established experi- 
mentally), but also that mathematics is entirely reducible 
to logic, and that intuition plays no part in it whatever. 

But Henri Poincare, who was described by Mr. Russell 
himself as " the most scientific man of his generation ", 
flouted the logistic contention. He denied that logistic 
(mathematical logic) gave any sort of proof of infallibility, 
or that it is even mathematically fruitful. It did certainly 
force us to say all that we commonly assume, and it forced 
us to advance step by step. But its labels are labels of con- 
sistency and do not in any way refer to objective truth. 
" The old logistic is dead." " True mathematics will continue 
to develop according to its own principles." " Fundamentally 
its development depends on intuition." 

Mr. Russell, in reply, said that mathematical logic was 
not " opposed to those quick flashes of insight in mathe- 
matical discovery " which Poincare " so admirably de- 
scribed ". Nevertheless, the main outlooks of the two men 
seem to be radically opposed. 

Mr. Russell has said elsewhere, " Mathematics is the 
science in which we do not know whether the things we 
talk about exist, nor whether the conclusions are true ". 
Apparently, then, Mr. Russell admits at least that logistic 
is not capable of discovering the mathematician's ultimate 
premisses, and is therefore not capable of establishing the 
truth of its final conclusions. It does, however, determine 



NATIVE GENIUS AND TRAINED CAPACITY 641 

the consistency of our conclusions with the premisses, and 
this is its undeniable merit. 



The Origin of New Mathematical Truths 

If ultimate mathematical truth is not discoverable by 
logistic, whence is its origin? Has it already an existence 
(as some contend) independent of us personally, something 
supra-sensible, already complete in itself, existing from the 
beginning of time, waiting to be discovered? is it of a priori 
origin? or is it actually created by mathematicians? 

The term a priori is ambiguous. Literally it signifies that 
the knowledge to which it applies is derived from something 
prior to it, i.e. is derivative, inferred, mediate. The metaphor 
involved in " prior " suggests an infinite series of premisses. 
But the term a priori is also often used to indicate that certain 
general truths come to the mind, to begin with, as heaven- 
born conceptions of universal validity, and are thus " prior " 
to all experience. Strictly, however, all a priori truths are 
derived truths. But derived from what? 

The mind seems to have a natural capacity for dictating 
the forms in which its particular experimental data may be 
combined. We may therefore correctly speak of the mind's 
creative powers, though not of its innate ideas. 

The mind's undoubted power of detecting identity and 
difference, co-existence and succession, seems to be original 
and inborn. Still, the power is exercised only on a contem- 
plation of actual things, from without or from within, and 
all such primitive judgments are individual. The mind 
compares two things and proclaims them to agree or disagree. 
The judgment is immediate, and it is felt to be necessary; 
it is irresistible and does not admit of doubt; it seems to be 
independent and to hang upon nothing else, and seems 
therefore to be primitive. But although the power is innate, 
this does not mean that the judgments themselves are innate. 

As primitive judgments are immediate, they are some- 
times described as intuitive. 

(E291) 42 



642 CRAFTSMANSHIP IN MATHEMATICS 



Intuition and Reasoning 

An intuition seems to be a general judgment immediately 
pronounced concerning facts perceived. But an intuitive 
judgment is as liable to error as is a reasoned judgment. 

There is a natural tendency to ascribe to intuition a 
peculiar authority, for it seems to confront us with an irre- 
sistible force foreign to the products of voluntary and 
reflective experience. But knowledge derived from intuition 
is as much experiential knowledge as directly conscious 
knowledge, and it is just as fallible. 

If we put on one side our purely primitive judgments, 
it seems very probable that, fundamentally, intuition and 
reasoning are identical, the former being instantaneous, the 
latter involving the notion of succession or progress. The 
difference then would be merely difference of time, every 
judgment of the mind being preceded by a process of reason- 
ing, whether the individual is able to recollect it or not. 

There are times when a great new truth suddenly comes 
to the mind of a mathematician. The combination of factors 
contributing to it seem to be a garnered knowledge derived 
from accumulated experience, a complete analysis of the 
given, a conscious connected reasoning, a systematic method 
of working, a natural capacity, and, finally, a flash of intuition. 
At some particular moment, the new truth flashes upon the 
vision as if light from all the other contributing factors was 
suddenly focused on the same point. 

The Limitations of the Teacher's Work 

Does not something of the same kind happen on a small 
scale when an intelligent schoolboy is solving a difficult 
problem? All ordinary methods of systematic attack may have 
failed him, yet light suddenly comes. Whence? Who shall 
say? Something from the rules of logistic, doubtless; some- 
thing from the boy's store of mathematical knowledge; 



NATIVE GENIUS AND TRAINED CAPACITY 643 

something from the boy's power of analysis of data; but 
most of all from the boy's own native capacity. It has been 
suggested that the truth suddenly emerges from a chance 
combination of the boy's data. We may put it that way if 
we like, but the " chance " seems to be very much more 
than an affair of mere randomness. 

If the boy's own native capacity is small, will the light 
appear? Can skilful teaching make up for native deficiency? 
In a considerable measure, yes; in a large measure, no. 
I do not think that great mathematical skill can ever be 
acquired by a boy with little natural mathematical endow- 
ment. We may meet with a considerable measure of sucfcess 
when we teach mathematics to average boys, but we shall 
never succeed in making such boys mathematicians. The 
boy whose average mark is 60 per cent is probably greatly 
indebted to his teacher. The boy whose average mark is 
90 per cent probably owes very much more to nature. 

Who can draw the line between the work of the boy's 
mind and what is skilfully presented to his mind to work 
upon? As teachers we have to teach methods, we have to 
teach analysis, we have to teach logic, we have to provide 
different types of mathematical knowledge. But can we do 
more? Can we increase the boy's own native mathematical 
capacity? Competent opinion answers that the old proverb 
of the silk purse still holds good. 



CHAPTER XLVIII 

The Great Mathematicians of History 

Few boys know anything about the great mathematicians 
except by name. " No time," says the mathematical master, 
a statement that cannot be denied. Yet, in order to include 



644 CRAFTSMANSHIP IN MATHEMATICS 

at least a little of the history of mathematics, I would sacrifice 
something else. Let a few of the really great mathematicians 
live once more, and let them be presented to the boys not 
merely as mathematicians but as human personalities. If 
it is to be but a few, who are the few to be? Personally I 
would select, from the ancients, Archimedes and Pythagoras; 
from Englishmen, Newton and Cayley; from Scotsmen, 
Napier and Madaurin; from Irishmen, Hamilton; from 
Frenchmen, Descartes and Pascal; from Germans, Gauss and 
Leibniz. Others will plead for the inclusion of Sylvester, 
Clerk Maxwell, Poincare, Einstein, and many more. Very 
well: the more the better. Let those selected come back 
and live their lives over again, and tell us exactly how they 
came to make their great discoveries. Let the boys know 
something of each great mathematical discovery, of each 
mathematical leap forward, and of the men responsible. 

One of the most readable books on the subject is Sir 
Thomas Heath's A History of Greek Mathematics, but of 
course it deals only with one period. It may be supplemented 
by Mr. Rouse Ball's A Short Account of the History of Mathe- 
matics, D. E. Smith's History of Mathematics, Professor 
TurnbulPs The Great Mathematicians, and F. Cajori's A 
History of Mathematics. 



CHAPTER XLIX 

Mathematics for Girls 

Opinions still differ about the relative mathematical 
ability of boys and girls. My own conclusions, derived from 
observations extending over a long period, are these, though 
I express them with some reserve: 

1. That a very small minority of girls in an average 



MATHEMATICS FOR GIRLS 645 

Form are as able, mathematically, as the ablest boys in the 
corresponding Forms in boys' schools, perhaps 3 per cent 
of girls against 10 per cent of boys. 

2. That the average mathematical ability of a Form of 
girls is rather lower than that of the corresponding Form of 
boys. 

3. That the interest of girls in mathematics is decidedly 
less than that of boys. 

4. That all girls should be compelled to take mathematics 
for 2 years (say, 11 to 13), but at the end of this time girls 
making no useful progress should give up the subject, arith- 
metic excepted, though the arithmetic should contain a 
certain amount of quite informal algebra and geometry, 
mainly by way of mensuration and the free use of formulae. 
Thus, in a large school with three or four parallel Forms at 
each stage from 13+ to 16+, the bottom Form would, as 
a rule, be a non-mathematical Form. 

However, the question is a woman's question, I would 
even say a question for women teachers generally rather than 
for mathematical mistresses only. Men are not likely to be 
able to consider adequately all the factors involved, or, 
indeed, to know them; and I am doubtful if a committee 
consisting only of mathematical mistresses would find it 
easy to free themselves entirely from the prejudice which 
naturally attaches to one's own subject. A committee for 
discussing the question should include a proportion of 
non-mathematical mistresses, but only those who could 
discuss the question objectively. 

The Girls' School Committee of the Mathematical 
Association issued a special Report on Mathematics in Girls' 
Schools in 1916. The Report was revised in 1928, and 
reissued in 1929. The Report is full of valuable suggestions, 
and should be read by all who are engaged in teaching mathe- 
matics to girls. There are, however, a few very able mathe- 
matical mistresses whose attitude towards the Report is a 
little critical. 



64:6 CRAFTSMANSHIP IN MATHEMATICS 

CHAPTER L 

The School Mathematical Library 
and Equipment 

The Library 



In the 134th number of ^^M^hermtical Gazett^l Dr. 
W. P. Milne writes a particularly interesting and suggestive 
article on this subject. A few paragraphs may be quoted: 

" There is nothing more extraordinary in the educational 
world at the present day than the change of attitude amongst 
teachers towards the subject they have to teach. Teachers of 
bygone ages were to a large extent content to regard their 
pupils as so many buckets into which they were content 
to pour a prescribed amount of intellectual material. In 
most cases the teacher himself did not know the sources 
whence came his stock-in-trade. Compare the attitude of 
teacKers~bf the present day, which is consciously or uncon- 
sciously, explicitly or implicitly, utterly different. For the 
teachers of to-day, their subject is not a finished structure; 
it is an organic growth, ever growing, ever changing, ever 
bein^ added to, ever having new methods devised and old 
cast aside. 

" Tr T do not think we quite realize what we owe to the 
Technical Colleges in getting rid of the old view. For the 
students of Technical Colleges, mathematics is essentially 
an^ instrument. They are willing to take much upon trust. 
They are afways on the look-out to use what they have 
learned, as a weapon. The result is inevitable. Mathematics 
acquired under these conditions may be imperfect, may be 
rough-and-ready, but the subject is at all events alive; it 
is confined within no rigid barriers, and is eternally moving 
and growing and changing. In the Technical College class- 
room, the propositions of geometry do not pass before the 



MATHEMATICAL LIBRARY AND EQUIPMENT 647 

audience in stiff and stately procession as the actors on the 
Greek stage; they are rather as the people of the market- 
place, hot and throbbing with life. Another great and in- 
calculable influence in helping to overthrow the old rigid 
and detailed view of a subject lies in the fact that men trained 
on research lines are slowly percolating through the staffs 
of the schools, and their attitude as teachers reacts on the 
taught. The combined results of these movements and 
tendencies is that both master and pupil recognize that it is 
possible to know only a very little about this ever-growing 
ever-changing subject, but what knowledge they do possess 
fills them with an eagerness to ask for more. There is in 
fact a widespread feeling among teachers of mathematics 
that every school ought to possess an up-to-date mathematical 
library, so that both teachers and taught may keep abreast 
of the times in their school work, and also catch a glimpse 
of the great regions of mathematical thought that lies 
ahead. 

" The school library should consist of two parts, one 
for scholars and one for masters, though these two should 
not be mutually exclusive. In general tone arid constitution 
the masters' portion should be altogether heavier and more 
serious than the scholars', because the teachers are older, 
more experienced, and have greater width of knowledge and 
intellectual power than the scholars. On the other hand, 
the senior division of the library is not intended for highly 
specialized experts in various branches of mathematics; the 
books should present rather * First Courses ', as they are 
popularly called, in the various subjects in which they deal. 
What schoolmaster does not want to know something more 
about that rapidly extending subject of Nomography? What 
modern teacher of mathematics does not want to know 
something about the theory of gunnery, submarines, stability 
of aeroplanes? On the other hand, the scholars' library should 
be more suited to^ their age and cajpacity; mathematical 
recreations and puzzles;" Rouse Ball's Jlistory of Mathematics; 
elementary history and theory of astronomy; the recent 



648 CRAFTSMANSHIP IN MATHEMATICS 

book on Astronomy by Professor R. A. Gregory; Pioneers 
in Science, by Sir Oliver Lodge; and so forth. 

" One cannot and ought not to lay down hard and fast 
rules about the composition of the school library. A con- 
venient method, however, would be to collect and catalogue 
the books under the headings of : (1) Biography, (2) History, 
(3) Philosophy, (4) Mathematical Analysis, (5) Geometry, 
(6) Applied Mathematics, (7) General. Such a composition 
gives plenty of scope for variety and elasticity." 

The whole article should be read by all mathematical 
teachers. 

In 1926, the Mathematical Association issued a.^ List of 

oks^ " likely to prove useful for reference to teachers and 
their more advanced scholars ". There are about JLOQ m a ^ 
mostly selected standard works, tried and proved, and re- 
commended for their real worth. No school can do better 
than begin with these, and add others from time to time. 
It should not be forgotten that the French and the Germans 
are much keener mathematicians than we are, and that some 
of their standard works in mathematics rank as classics. 

Those unfamiliar with German may be interested to know 
that enough about the language may easily be mastered in 
a month for any technical German work to be read with ease. 
(This does not of course apply to the speaking or writing of 
German!) 

At the end of the Report on Elementary Mathematics in 
Girls' Schools (Mathematical Association) are some useful 
appendices: 

Appendix C. Lists of Books for the Libraries of Girls' 
Schools (an admirable selection). 

Appendix D. List of some of the Articles from the 
Gazette that are of general interest to teachers. (Many others 
may be found by referring to back numbers of the Gazette.) 

Appendix E. List of Reports published by the Mathe- 
matical Association. 

A bound copy of the Gazette should be added every year 
to the teachers' section of the school library. The articles 



MATHEMATICAL LIBRARY AND EQUIPMENT 649 

are almost always helpful to the practical teacher, and every 
copy of the Gazette is full of useful hints. 

The teachers' section of the mathematical library should 
also contain the Board of Education's Special Reports on 
Educational Subjects, numbers 12, 13, 15, 16, 18-34, all 
dealing with mathematics in some form or other, all written 
by well-known experts. 

Mathematical teachers may be reminded that, whilst 
their school libraries are in the making (or at any other time), 
they may, if they are members of the Mathematical Association, 
borrow books from the Association's own library, which is 
under the care of Professor Neville at Reading. A list of 
books in this library is published by the Association. 

Mathematical teachers should keep an eye on the columns 
of the Mathematical Gazette, Nature, The Educational Outlook 
and Educational Times, The Times Educational Supplement, and 
The Journal of Education, for reviews of new books. 

Equipment 

Schools provided with a separate Mechanical Laboratory 
are usually well stocked with apparatus for teaching mechanics 
and practical mathematics, including such things as screw 
gauges, calipers, verniers, spherometers, opisometers, slide 
rules, a planimeter, cycloidal curve tracers, an ellipsograph, 
link-motion apparatus, a binomial cube, a variety of wooden 
and wire geometrical models, and models to illustrate pro- 
jection. When there is a special mathematical laboratory, 
the room is usually given up to the Lower and Middle Forms, 
and few instruments of precision are to be found in it. Indeed 
these are generally too expensive to be purchased, but they 
can always be seen and examined at the Science Museum at 
South Kensington. 

Not all mathematical teachers are aware of the fine collection 
of instruments in the mathematical section of the Science 
Museum. I have often seen classes of boys under the 
guidance of a teacher, examining the instruments, apparatus, 



650 CRAFTSMANSHIP IN MATHEMATICS 

and working models in the physics, mechanical, and engineer- 
ing sections of the Museum, but the mathematics and the 
geodesy and surveying sections are usually deserted. Amongst 
the things that all boys should see in these sections are the 
following: 

1. Instruments for drawing curves: trammels, ellipso- 
graphs, parabolographs, &c. 

2. Ellipsoids, hyperboloids, paraboloids: various models 
of curved surfaces formed by (i) intersecting layers of stiff 
paper, (ii) series of stretched strings. (Some of these are 
singularly attractive and wonderfully instructive. I remember 
as a youth spending my very scanty supplies of pocket-money 
for several months in purchasing materials and tools for 
reproducing some of these.) Pairs of intersecting cylinders 
and of cones. Roof structures. Skew bridges. 

3. Instruments for preparing perspective drawings from 
plans and elevations. 

4. Sundials and clocks 

5. Calculating machines: Pascal's, Morland's, Stanhope's, 
the Brunsviga. 

6. Difference and analytical engines: Babbage's, Scheutz's. 

7. Slide rules: straight, circular, spiral, cylindrical, and 
gridiron types. 

8. Instruments for solving equations. 

9. Linear integrators, planimeters, integrometers, inte- 
graphs. Professor Boys' curve - drawing integrator, and 
his model of a polar planimeter. 

10. Harmonic analysers and integrators. 

11. Jevons' logical machine. 

12. Rangefinders and tacheometers. 

13. Clinometers, azimuth compasses, prismatic com- 
passes, early theodolites, altazimuth theodolites, geodetic 
theodolites, zenith sectors and telescopes. 

14. Mine surveying and marine surveying instruments. 

15. Air survey maps, geodetic triangulation maps, &c., &c. 

The services of a well-informed museum guide may be 



MATHEMATICAL LIBRARY AND EQUIPMENT 651 

obtained for the asking; he will accompany visitors on their 
rounds, and explain the instruments under examination. 

Many of the machinery models in the Science Museum 
are in motion, being driven by compressed air. Visiting 
boys should always be taken to see them. 

Visits may also profitably be made to such places as 
Cussons' Technical Works, Manchester; to Sir Howard 
Grubb, Parsons, & Co.'s optical works at Newcastle; to 
Hilger's; and to some of the better-known instrument makers. 

Teachers interested in practical measuring might read 
Mr, F. H. Rolt's Gauges and Fine Measurements, the standard 
work on measuring machines, instruments, and methods. 

I am doubtful if any part of the mathematical equipment, 
even instruments of historical interest, should be kept in the 
school museum. The school museum is apt to be a holy 
place, to be visited seldom, and then silently lest the dust 
should be disturbed. 



APPENDIX I 



A QUESTIONNAIRE FOR YOUNG 
MATHEMATICAL TEACHERS 

1. In teaching arithmetic, to what extent is it advisable to give 
a logical justification of the rules you teach at the time you teach 
them? Illustrate your answer by reference to division of vulgar frac- 
tions, and to multiplication of decimals. 

2. What is your general plan for teaching the tables to young 
children? Doubtless you are convinced of the necessity for plenty of 
ding-dong work to make the tables perfect. What is the best means 
of dealing with a visitor who suggests that such work is unintelligent? 

3. Draft a few notes suitable for a Student Teacher, instructing 
him how to proceed when teaching subtraction to beginners. 

4. What is the proper function of so-called " mental " arithmetic? 
Discuss the merits of (a) oral work, and (6) practice on paper, in 
mathematical teaching, and assign to each type its proper place in 
the different Forms of the school. 

5. Where in the school would you begin algebra, and how? Out- 
line a suitable course of work for the first two years, and show how 
you would link it up with the mensuration, arithmetic, and geometry. 

6. A slow-witted boy of Form IV tells you that he does " not 
quite see " how the product of V3 and V5 can be V 15. On question- 
ing him you find that he has failed to grasp the essentials of the lesson 
you have just given. Sketch out a new and very simple lesson to meet 
the case. 

7. Would you include the theory of annuities in a school mathe- 
matical course? If so, on educational grounds or on utilitarian grounds? 

8. What is your experience of teaching logarithms to bottom Sets? 
Is the practice desirable? If so, on what grounds? 

9. The pedagogical treatment of parallel lines is admittedly very 
difficult, and no teacher would suggest that a rigorous treatment at 
the outset is possible. How would you make the subject more and 
more logically exacting in the successive Forms from II to VI? 

653 



654 CRAFTSMANSHIP IN MATHEMATICS 

10. Defend the principle that formal definitions in geometry 
should never be provided ready-made by the teacher. 

11. Criticize the value of the work on graphs commonly done in 
schools. How would you modify such work? 

12. In what part of the school mathematical course would you 
first introduce the idea of incommensurables? Should it be included 
at all in a school with a leaving age of 16? If so, justify your opinion. 

13. Sketch out a course of numerical trigonometry for D Sets 
(Forms III to V), introducing as much field work as possible and 
reducing academic work to a minimum. 

14. Whose business is it to teach mechanics, the teacher of mathe- 
matics or the teacher of physics? If the former, how is he to proceed 
if he has had no training as an engineer? If the latter, how is the mathe- 
matical side of the subject to be dealt with effectively? 

15. In a Sixth Form course of mechanics, how far is it (a) possible, 
(b) desirable, to replace some of the elegant but rather academic and 
useless problems in ordinary dynamics by an elementary course in 
the dynamics of astronomy and geology? 

16. Kepler 's Third Law is found to hold good for the earth as 
well as for the other planets. How would you demonstrate to a Sixth 
Form that this fact alone affords strong evidence that the earth itself 
is a planet? 

17. Do you find that school mathematics has improved since the 
calculus has been taken up in Forms below the Sixth? Defend (a) its 
inclusion in, (b) its exclusion from, such Forms. What do you con- 
sider to be the educational gain of a course of analysis in school work 
for boys not going on to the University? 

18. Mathematicians have investigated many geometrical curves 
that have little relation to practical life. On the other hand, they 
have almost neglected to investigate the curvature of such common 
things as eggs. The reason sometimes put forward is that no two 
exactly similar eggs have ever been discovered. Assuming this to be 
a fact, does the fact justify the neglect? If it does, what have you to 
say about the mathematics of biology generally? 

19. The physiological exchange which is inseparable from active 
life is conducted through limiting surfaces, external and internal. 
Provided the form remains unchanged, the bulk of a growing cell, 
tissue, or organ increases as the cube of the linear dimensions, the 
surface only as the square. Accordingly, as growth proceeds, the 
proportion of surface to bulk decreases, until a point of physiological 
inefficiency is reached. Your biological colleague, who is not a 
mathematician, appeals to you for help over this difficult question 
of inter-relation between growth and form, whether applied to ex*- 
ternal surfaces or to internal conducting tracts in plants and animals. 
Sketch out a course of lessons dealing with the mathematics of this 
biological " size-factor ". 

20. Sixth Form work on (a) Capillarity and (b) Viscosity is almost 



APPENDIX 655 

always treated mathematically in the main, experiment playing a 
very small part. Assuming that this treatment is correct, it is clearly 
the business of the mathematical master to be responsible for it. 
Give an outline of three or four lessons in each subject, and say 
exactly what help, and where, you would expect from your Physics 
colleague. 

21. A Sixth Form boy tells you his Physics teacher has said that 
Van der Waals' equation is not sufficiently representative of the facts, 
and he asks you if you will explain " the mathematical fallacy in the 
equation ". Devise a suitable reply. 

22. Write out brief notes of a lesson on Lissajous figures, includ- 
ing a demonstration of the method of establishing the general equations 

- = sinraO, ^ sin(w6 + p), where m and n are proportional to the 
a b 

frequencies of the horizontal and vertical vibrations. Should the 
mathematics or the physics master demonstrate the fact that the 
gradual changes from one figure to another depend on the gradual 
change of p? Why? 

23. Have you found School Certificate examination requirements 
clash with your own thought-out plans for mathematical teaching? 
If so, how? Are you quite sure that the clashing is real? If so, take a 
recent School Certificate paper, modify it in accordance with your 
own views, and see if your colleagues agree. 

24. Here are two well-known old problems, much too difficult for 
the average boy: 

(i) A spider at one corner of a semicircular pane of glass 
gives chase to a fly moving along the curve before him, the fly 
being 30 ahead when the chase begins. Each moves with its own 
uniform speed. The spider catches the fly at the opposite corner. 
(1) Trace the spider's path; (2) determine the ratio of the speeds 
of the two insects. 

(ii) A horse is tethered to a stake in the hedge of a circular 
field, the rope being just long enough to enable him to graze over 
half the area of the field. Determine the length of the rope in terms 
of the diameter of the field. 

Is it worth while spending time over such examples of the type of 
(i) the first, (ii) the second? Discuss examination conundrums generally 
and their injurious effect on rational school practice. 

25. Young mathematical teachers generally try to adopt methods 
which, in exposition, shall be mathematically flawless. Have you 
ever known " intelligent " teaching to be productive of unintelligent 
reception? If so, how do you explain it? Who is likely to be the more 
to blame, the teacher or the boy? 

26. What do you understand by a proof! Criticize the term as 
commonly used in mathematical teaching. At what stage do you 
advise the introduction of simple proofs? When would you introduce 



656 CRAFTSMANSHIP IN MATHEMATICS 

formal proofs of the fundamental theorems of geometry (congruent 
triangles, angles at a point, parallels, &c.)? What is your opinion of 
the value, to boys, of the usual book proofs of H.C.F. (algebra) and 
the Binomial Theorem? 

27. What do you consider to be the essential distinction between 
" pure " and " applied " mathematics? A chalk or a plumbago line 
drawn with the compasses is as concrete a thing as the iron rim to a 
flywheel: are you therefore doing " purer " work in the classroom 
than in the laboratory? How do you defend the use of the term 
" pure " in any of your mathematical teaching except perhaps in the 
work of those Sixth Form specialists who are capable of grasping the 
A B C of mathematical philosophy? 

28. How do you account for so few persons being interested in 
mathematics, although the great majority of educated people must 
have done a fair amount of mathematics at school? How would you 
modify the present course of school mathematics in order to ensure 
a greater permanent interest in the subject on the part of learners? 

29. Plan out a course of work for the Sixth Form, specialists and 
non-specialists alike, calculated to develop and maintain a life-long 
and intelligent interest in mathematics, the course not to be so recre- 
ational that rigour is seriously sacrificed. 

30. What commonly taught topics would you advise should be 
deleted from a school course? Why? What others would you substi- 
tute? 

31. A boy asks the question, " Exactly when did the twentieth 
century begin?'* How would you answer it? How would you deal 
with his meridian difficulty, especially if he lived on the 180 meridian? 
How would you make this difficulty a good jumping-ofT place for 
introducing first notions of Relativity? 

32. Devise courses of instruction for: 

(i) Junior Elementary Schools, 8+ to 11 + . 
(ii) Senior Elementary Schools, 11+ to 14-f-. 
(iii) Central Elementary Schools, 11+ to 15 + . 
(iv) Preparatory Schools, 8+ to 13 + . 
(v) Secondary Schools, 11+ to 18 + . 

In the last case assume that the five Forms Upper III, Lower and 
Upper IV, Lower and Upper V, each consists of four graded Sets, 
A, B, C, and D. Show how the work of the Sets should differ in (i) 
content, (ii) ^treatment. 

33. Your school is of sufficient importance to receive each year 
from a leading Training College a few students who have taken good 
degrees in mathematics, and you are entrusted with the purely prac- 
tical side of their professional training. What measures would you 
adopt to make such training effective? 

34. It is sometimes said that a classical training is " necessarily " 
productive of a much greater appreciation of the beautiful than is a 
mathematical training; that a real appreciation of a beautiful thing 



APPENDIX 



657 



is always accompanied by a glow, whereas a mathematician's apprecia- 
tion is always reasoned and therefore cold. The contention is, of 
course, silly, but how would you formulate a defence that would 
carry conviction to an opponents mind? 

35. You are appointed chief mathematical master of a new school 
where a room 36 ft. by 25 ft. has been set aside for a mathematical 
laboratory. Give full details of the fittings and equipment you would 
provide for elementary and for advanced work, the expenditure to be 
limited to 500. 

36. What is the proper function of school textbooks in mathe- 
matics, apart from the exercises they provide? Distinguish between 
textbooks for Lower Forms and those for Higher. 



APPENDIX II 



NOTE ON AXES NOTATION 

In drawing graphs I have followed the usual notation. But the 
following figures indicate a useful reform: 




Fig. i 



1 . Mark the x and y axes merely by arrows pointing in the positive 
direction. (Figs. 1 and 2.) 

( K 291 ) 43 a 



658 CRAFTSMANSHIP IN MATHEMATICS 

2. Indicate dimensions by means of single-headed lines instead of 
the draughtsman's usual double-headed lines, the direction of the 

arrow indicating the actual direc- 
tion in which the line is to be 
measured. For instance, the 
length QR is +(x + 1J), and the 
length RP is -\-(y 1); the latter 
is the difference in the dimen- 
sions measured in the positive 
direction. (Fig. 1.) 

3. Do not use a dimensional 

line for length r, since by con- 
vention a radius vector is always 
taken with the positive sign. 
(Fig. 1.) 

4. For angles, use -a single-headed (not a double-headed) arrow, 
and let the arrowhead indicate the direction of rotation. (Fig. 2.) 



O 




INDEX 



Acceleration, 428. 

Accuracy, 18, 28. 

Adams, 20. 

Addition. 23. 

Advanced mathematics, 553. 

Algebra, addition and subtraction, 127. 

and arithmetic in parallel, 132. 

beginnings, 122. 

cyclic expressions, 202. 

formal beginnings, 125. 

geometrical illustrations, 134. 

links with arithmetic and geometry, 

132. 

multiplication, 130. 

product distribution, 190. 

2 notation, 193. 
Algebraic equations, 205. 
roots of, 205. 

roots of biquadratics, 208. 

roots of cubics, 208. 

roots of quadratics, " either . . . 

or ", 205. 

expansions, 195. 

expressions, complex forms, 194. 

factors, 177. 

geometrical models for, 179. 

identities to be learnt, 204. 

manipulation, 177. 

phraseology, 181. 

signs, 125. 

symmetry, 201. 
Amicable numbers, 597. 
Andrews, 607. 

Analysis of algebraic problems, 215. 

of curves, 465. 

of geometrical riders, 319. 
Animal locomotion, 585. 
Apothecaries' weight, 49. 
Applied mathematics, 489, 634. 
Approximation methods, 412. 
A priori truths, 641. 
Archimedes, 480, 486, 644. 

spiral, 571. 

Areas and volumes, 50. 
Aristotle, 625, 639. 
Arithmetic, 23. 

commercial, 109, 117. 

( E 291 ) 



Arnoux, 607. 

Art, proportion and symmetry in, 558, 

58